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Topologies on Spaces of Subsets Ernest Michael Transactions of the American Mathematical Society, Vol. 71, No. 1. (Jul., 1951), pp. 152-182. Stable URL: http://links.jstor.org/sici?sici=0002-9947%28195107%2971%3A1%3C152%3ATOSOS%3E2.0.CO%3B2-4 Transactions of the American Mathematical Society is currently published by American Mathematical Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/ams.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected] http://www.jstor.org Fri Jul 27 08:18:31 2007 TOPOLOGIES ON SPACES OF SUBSETS BY E R N E S T MICHAEL 1. Introduction. In the first part of this paper ( § § 2 - j ) ,we shall study various topologies on the collection of nonempty closed subsets of a topological space(l). In the second part ($86-8), we shall apply these topologies to such topics as multi-valued functions and the linear ordering of a topological space. T o facilitate our discussion, we begin by listing some of our principal conventions and notations : Convention 1.1. Let X be the "base" space. Then 1.1.1. An element of X will be denoted by lower case italic letters (for example, x). 1.1.2. A subclass of X will be called a set, and will be denoted by upper case italic letters (for example, E ) . 1.1.3. A class of sets will be called a collection, and will be denoted by light-face German letters (for example, 93). 1.1.4. A class of collections will be called a family, and will be denoted by bold face German letters (for example, '3 ). Convention 1.2. By a neighborhood of a class, we shall always mean a neighborhood of this class considered as an element of a topological space, not as a subclass of such a space. Notation 1.3. 1.3.1. A topological space X , with topology T, will be denoted by (X, T). 1.3.2. A uniform space X , with uniform structure U, will be denoted by . [X, u]; the topology which U induces on X will be denoted by Notation 1.4. Let (X, T ) be a topological space. Then d ( X ) = ( ECXI E is not empty f , 2X = ( E cx E is closed and not empty f , J,(X) = { ~ € 2 E~ has 1 a t most n elements f , J(X) = E is finite], e ( X ) = ( E € 2 E~ is compact f (2). The following notation will be useful for defining and discussing our topologies : Notation 1.5. If ( U, f is a collection of subsets of a topological space X , I UI I {EE~XI I Presented to the Society, April 29, 1950; received b y the editors April 26, 1950 and, in revised form, December 18, 1950. (I) Such collections are often called hyperspaces. (2) Strictly speaking, we should write T ) and so forth, since e(X) depends on T. Where no confusion can occur, we shall, however, adopt the simplified notation. C(X, TOPOLOGIES ON SPACES O F SUBSETS 153 . then (UI),EI (or (Ul, . . . , U,) if I = ( 1 , . , n } ) denotes ( ~ € 2 ~ 1 ~ C U I E ~ U 1and , E A U l # @(3) for all LEI}. T h e first step towards topologizing a collection of subsets of a topological space X was taken by Hausdorff [lo, §28](4),who defined a metric on 2X in the case where X is a bounded metric space(5). If we extend the Hausdorff metric to d ( X ) , then sets with the same closure are a t zero distance from each other; hence the extended distance function is not a metric. When topologizing the subsets of a more general topological space, the situation is quite similar: On 2X, these topologies have decent separation properties, but when extended to d ( X ) , they fail to be Tl (in fact, for all E C X , any neighborhood of ?? contains E ) . We shall therefore, except in $5, restrict our considerations t o 2X. One of the features of the Hausdorff metric is that the function('j) i : X -+ 2X, which maps x into { x } ( 3 ) , is an isometry. In order to retain analogous properties for more general X , it is necessary, first of all, that the one-point subsets of X be elements of 2X; we shall therefore assume throughout this paper (except in 4.9.1. and 4.9.2) that the base space X is TI. We now call a (topology/uniform structure/metric) on 2X admissible with respect to a (topology/uniform structure/metric) on X if the function i is a (homeomorphism/uniform isomorphisn~/isometry). I t is reasonable to require that all structures on 2X be admissible, and the ones that we are about t o define do, in fact, satisfy this requirement. Guided by the above preliminary observations, we are now ready t o topologize 2X when X is not a bounded metric space. For uniform spaces, we follow Bourbaki [ I , p. 96, Ex. 71 in making the following definition. DEFINITION1.6. Let [x,U] be a uniform space with index set A. For each E . then define the uniform structure 2U on E E 2 X , let 8,(E) = ( V , ( X ) ) ~ ~ We 2X to be the one generated by A and the collection %,(E), and we call 1 2 ~ 1 the uniform topology. The verification that this actually generates a T1 uniform structure on 2X is tedious, but entirely straightforward; it is given as an exercise in [I, p. 97, Ex. 71, and will be omitted here. For an arbitrary topological space, we follow Vietoris [14, p. 259, M - ~ m ~ e b u n ~in e ndefining ] a topology on 2X as follows: DEFINITION1.7. If (X, T) is a topological space, then the $nite topology 2T on 2X is the one generated by open collections of the form (Ul, . . , U,) , U, open subsets of X . with Ul, (or C1 (E) if the expression for E is long) (3) We use ,Ef to denote the null set. We also use to denote the closure of a set E , and E ' (or X - E ) to denote the complement of E (in X). If x is a n element of X, we use { x } to denote the subset of X whose only element is x. Similarly for subclasses or elements of hyperspaces. (4) Numbers in brackets refer to the bibliography a t the end of the paper. (6) For a n extensive study of this metric topology on ZX for metric compact X, see Kelley [I 11. (6) The terms function, map, and mapping are used synonymously in this paper, and do not imply continuity. 154 ERNEST MICHAEL [July I t is easy to see t h a t if X is a bounded metric space, then 2U agrees with the Hausdorff metric. The finite topology 2T, on the other hand, agrees with this metric only if X is compact (see Proposition 3.5); this is no calamity, however, for in some important respects the finite topology behaves much better than the metric topology (see 2.4.1 and 4.10). $2 deals with fundamental properties of the topologies on 2X. What is probably the most interesting result of this section (Theorem 2.5) can be paraphrased as follo~vs:X compact union of closed sets is closed, and a compact union of compact sets is compact. $3 deals with relationships between the various topologies on 2". T h e most important results are that, for a uniform space [ X , li], the uniform and finite topologies agree on e ( X ) (Theorem 3.3), and t h a t , for a normal X and a special U, they even agree on 2X (Theorem 3.4). In $4, we investigate what properties of X are carried over to 2" or to C ( X ) . Among the former are compactness (Theorem 4.2), connectedness (Theorem 4.10), and most separation properties in somewhat iveakened form (Theorems 4.9.1-4.9.7). As for e ( X ) , it appears to inherit intact virtually every property of X. In addition to the properties just mentioned for 2", e ( X ) inherits the separation properties in undiluted form (Theorems 4.9.84.9.13), local compactness (Proposition 4.4), and local connectedness (Theorem 4.12). In $5, we begin by extending our topologies to d ( X ) (Definition 5 . l ) , and showing t h a t the function(@which maps a subset of X into its closure is a retraction (Theorem 5.3). Next we define certain functions among the hyperspaces (Definition 5.5), and derive their properties (Corollary 5.6-Corollary 5.8). Typical among these is t h a t the function a : d ( d ( X ) ) + d ( X ) , which maps a collection of sets into its union, is continuous (Theorems 5.7.1 and 5.7.2). S e x t we study the relationships between a function f : X -+ Y and the function it induces among the hyperspaces (Theorem 5.10). We conclude this section by defining a new concept, the saturate of a collection (Definition 5.13), and deriving its properties. &Aninteresting result is t h a t the union of a closed and saturated collection of closed sets is closed (Corollary 5.16). Before continuing this summary, we define a concept which plays a crucial role in the investigations of the remaining sections. DEFINITION1.8. Let GE2". -4 function f : G + X is called a selection if it is continuous, and if f ( E ) EE for all E E G . In $6 we discuss the following question: Given a function g: Y + 2", when is it possible to find a continuous g': Y -+ X such t h a t gl(y)Eg(y) for all y E Y? We attack this problem by dividing it into t ~ v oparts; the first part deals with the continuity of g, while the second deals with the properties of X alone. Among the new questions thus raised, we have the following: Question 1. For what X does there exist a selection from 2X (or from e ( X ) ) to X ? 19511 TOPOLOGIES ON SPACES O F SUBSETS 3 55 Question 2. For what X does there exist, for ever)- disjoint covering TI of X by closed (compact) sets, a selection from to X ? In $ 7 we study Question 1 and related topics. Question 1 itself is completely answered by the following theorem. THEOREM 1.9. A m o n g the following statements, 1.9.2 + 1.9.1 and 1.9.4 for a n y Hausdorff space ( X , T ) . I f , moreover, every connected component of X i s open, then conversely, 1.9.1 + 1.9.2 a n d 1.9.3 + 1.9.4: 1.9.1. There exists a selection f : 2" + X . 1.9.2. There exists a linear ordering o n X such that the order topology i s coarser t h a n T , and such that every closed ( T )set has a j i r s t element. 1.9.3. There exists a selection f : e ( X ) + X . 1.9.4. There exists a linear ordering o n X such that the order topology i s coarser t h a n T . t 1.9.3 We note t h a t i f , in particular, X i s a connected Hausdorff space, then 1.9.1 F! 1.9.2 and 1.9.3 F! 1.9.4. These equivalences should be compared with [6, Theorem 11, where another necessary and sufficient condition for 1.9.4 is given. T h e chapter concludes with some examples of spaces which do, or do not, satisfy 1.9.2 or 1.9.4. In $8 we begin by showing t h a t certain selection properties are retained under mapping^(^) which are continuous, open, and closed. We conclude the section by giving some partial answers to Question 2. In the appendix, we begin by re-examining the foundations of the finite topology. This leads us to some new theorems, which in turn lead to new applications. We conclude by outlining a similar program for the uniform case. 2 . Fundamental properties of the topologies on 2". \Te begin this section by observing t h a t the topologies 2 T and 1 2 ~ 1the , uniform structure 2 u , and the Hausdorff metric are all admissible. \Ve now examine the finite topology. PROPOSITION 2.1(7). Th,e collections of the f o r m ( U l , . . . , U,), with Ul, . . . , U , open in X , form a basis for t h e j i n i t e topology o n 2". Proof. (a) 2 X = (X). U;, (b) Let U = ( l i l , . . . , U,), % = ( V l , . . . , V,,), and let U = U:-, . , li,A V, VIA U, . . . , VmAU). Q.E.D. V = UzlV;. Then U A % = ( U l A V, Our definition of 2 T is essentially the one given by Vietoris and Choquet. I t is easy to verify t h a t it is equivalent t o Frink's neighborhood topology (see [7, $ 1 2 1 ) . Before leaving this topic, we give another definition of 2 T , similar to Frink's, which makes this topology appear especially "natural." DEFINITION1.6a. I f ( X , T ) i s a topological space, we s a y that a topology o n (?) This assertion remains true if 2 X is replaced by ( )+). For the topology on see Definition 5.1. &(x), &(x) (and ( ), if occurring, by 156 ERNEST MICHAEL [July 2Xi s acceptable if in i t ( E € 2 ~ 1 E C A ) i s closed for all closed A C X , and i s open for all operz A C X . T h e n theJinite topology 2T i s the coarsest acceptable topology o n 2I. Note. T h e uniform topology is, in general, not acceptable. I t might be interesting to study the properties of an arbitrary acceptable topology, but this will not be done in this paper. The following lemmas deal with simple properties of the finite and uniform topologies: LEMMA2.2(7).Let (X, T ) be a topological space. T h e n : 2.2.1. { ~ € EC 2 A )~i s closed in (2X, 2T) i f A C X i s closed. 2.2.2. { E € 2 ~E A A #,@25) i s closed in (2X, 2T) i f A C X i s closed. I f [x, U] i s a u n i f o r m space, then 2.2.1'. ( ~ € 2E C~ A ) i s closed in (2X, 1 2 ~ 1 if ) A C X i s closed. 2.2.2'. ( ~ € 2E A~ A #,@) i s closed in (2X, 1 2 ~ 1 i)f A C X i s closed, provided A i s compact. Proof. T h e first three statements follow directly from the definition. T h e last statement follows from the fact that if A and B are closed subsets of a uniform space, one of which is compact, then there exists an index a such that V,(A)A V,(B) = ,@ (see Notation 3.1.2). Q.E.D. T h e following assertions follow immediately from the definitions, and are stated without proof: LEMMA2.3 (7). 2.3.1. ((Ui, . . . , U,)C(Vl, . . , V m ) )F! (U:=l U i c U E 1 Vi, a n d for every Vi there exists a Uj such that U j C Vi). 2.3.2. In thejinite topology o n 2X, C1 ((Ul, . . , U,)) =(Dl, . , U,)(3). 2.3.3. I f ( U,),€ A i s a basis for the neighborhoods of x in X , then ((U,)) ,€ A i s a basis for the neighborhoods of ( x ) in 2X in either the u n i f o r m or the $finite topology. - . PROPOSITION 2.4(7). I f (X, T) i s a topological space, t h e n : 2.4.1. J(X) i s dense in (2=, 2T). 2.4.2. I f X i s Hausdorff, then J,(X) i s closed in (2X, 2T) for all n 2 1. 2.4.3. T h e natural m a p pr: X n -+J,(X), dejined b y pr ((xl, . . , x,)) = {xi, . , X , ) , i s continuous. - Note. (a) If [X, U ] is a uniform space, then 2.4.2 and 2.4.3 also hold with the uniform topology on 2X (the mapping pr is now, in fact, uniformly continuous). Assertion 2.4.1 is, however, false in this case. (b) Assertion 2.4.3 is false for injinite products. (c) If X is not Hausdorff and if J,(X) #2X, then J,(X) is not dense in (2X, 2T). The following theorem generalizes the elementary fact that a finite union of closed (compact) sets is closed (compact). TOPOLOGIES ON SPACES OF SUBSETS 19511 THEOREM 2.5(8). 2.5.1. If (X, T) is a regular space, then 2.5.2. If (X, T) is a topological space, then 2.5.1'. If [x,U] is a uniform 2.5.2'. If [x,U] is a uniform space, then space, then Proof. 2.5.1. Suppose etEC(2X). Let A = UEE8E, and let ~ € 2Then . for each closed neighborhood N of X , ~ A ~ E EE~A x N #I @ ] is a closed subcollection of 8 (see 2.2.2), and the family of all such subcollections has the finite intersection property. Since % is compact, this family has a nonempty intersection D C 8 . But since X is regular, x is an element of every element of Dl whence x E A . 2.5.2. Suppose A = U E E 8 E , with 8 C e ( e ( X ) ) . Let U be a collection of open subsets of X which covers A. Now let E E 8 ; then E is a compact subset of X ,and hence there exists a finite subcollection f UE,I, . . . , U E , ~ ( E )) of U which covers El and all of whose elements intersect E . Hence, for each EE8, UE=(UE,l, . . . , UE,,(E,)is an open neighborhood of E, and therefore { U E ) E E 8 is a covering of et by open collections. But 8 is compact, so there exists a finite subcollection { E ~. ,. . , E m )of 23 such that { U E ~ ,. . . , UE,] is a covering of 8 . Hence, finally, f ~ ~ ~ , ~ ] : ~ : : : : : , ; ; l n ( ~ is i ) a finite subcollection of U which covers A. 2.5.1'. Suppose that 8 E C ( 2 X ) . Let A = U E E 8 E l and let xEA'. For each EE8, pick an index a ( E ) such that [email protected] V m ' ~ ) ( Ethen ) ; pick a P(E) E Z ( a ( E ) ) and a y ( E ) E Z ( P ( E ) ) (see Notation 3.1). Then f B - , ( ~ ) ( E ) ) Eis~a~covering of 8 by open collections, and hence there exists a finite subcovering { B Y ( E i ) ( ~ i ) Let y = max f y ( ~ i ].) I t follows easily that x @ V,(A), and hence that x has a neighborhood which is disjoint from A. 2.5.2'. This follows directly from 2.5.2 and Theorem 3.3 of the next section. Q.E.D. (8) For a stronger version of this assertio11, see the Appendix, 158 ERKEST M I C H A E L [July Theorem 2.5 and its extension in the appendix have many applications, where they reveal the basic similarity between some apparently unrelated problems. In particular, Theorem 2.5 yields especially simple and transparent proofs for the following theorems in group theory: ( 1 ) If G is a topological group, A a compact subset of G, and B a compact (closed) subset of G, then A B is compact (closed). ( 2 ) If G is a topological group, H a compact subgroup of G, q5 the canonical mapping of G onto G/H, and Q a compact subcollection of G/H, then q5-'(Q) is compact in G. T o prove the first theorem, let U be the right uniform structure on G. Then the function f : [G, U ] -+ [2G,2 U ] ,defined by f ( x ) = x B , is clearly uniE ,concluformly continuous. Hence f ( A ) is compact; since A B = U E ~ / ( G )the sion follo\vs fro111 2.5.2' (2.5.1'). T o prove the second theorem, we observe that d - l ( Q ) = UeEeE; the conclusion now follows from 2.5.2' and the first part of the follo\~-ingproposition. PROPOSITIOK 2.6. Let G be a topological group, H a closed, n o r m a l subgroup of G. I f U i s the right (left) u n i f o r m structure o n G, t h e n the right (left) u n i f o r m structure o n G I H i s the one induced o n G / H a s a subcollection of [2G, 2 U ] . Furthermore, G / H i s a closed subcollection of ( 2 G , 1 2 ~ )1f o r either U. Proof. The first assertion is [ 2 , p. 31, Ex. 51. Its verification, as well as that of the second assertion, is straightforward, and we omit it. Q.E.D. Proposition 2.6 leads us to consider the factor topology (topologie quotient in [ I , p. 5 2 , Definition I ] ) . In general, the following proposition, whose proof we omit, is all we can assert; in special cases we have better results, such as 2.6 above, 5.10.4, and 5.11.1. PROPOSITION 2 . 7 ( 8 ) . I f 3 i s a disjoint cooering of a topological space ( X , T ) b y closed subsets, t h e n the factor topology o n 2 i s coarser t h a n the relative topology znduced o n D b y 2". The follom-ing proposition is analogous to Theorem 2.5. The proof is elementary, and \ve omit it. 2 . 8 ( 7 )( 8 ) .I f 8 i s a collection of subsets of X which i s ( d i s j o i n t / a PROPOSITION subcollection of 2 X ) a n d connected in the (factor/$nite) topology, a n d ( a l l / o n e ) of whose elements are connected, t h e n U.gEsE i s connected. \\re collclude this section with some examples. In all cases, U will denote the natural uniform structure on X . E.~anzple2.9. Let X be the plane, and let 3 be the disjoint covering of X I\ hose elements are the horizontal lines in the open upper half-plane, and the vertical half-lines in the closed lower half-plane. Then 21U is discrete on D, 1 2 ~ 1is coarser on 52 but still disconnected, while the factor topology on D is coarser yet-and is connected. TOPOLOGIES ON SPACES OF SCBSE'TS 19511 159 Example 2.10. Let A be the graph of f ( s ) = l / s , Y the y-axis, X = A U Y, and the covering of X \\hose elements are A and the sets { x ) with z E k'. Then 2IC1 and the factor topology agree on 9, and are coarser there than 1 2 ~ 1; A is a n isolated element of 3 in the former topologies, b u t not in the latter. Example 2.11. Let X be the real line, 93 the collection of all closed (finite o r infinite) intervals. Then X is a n isolated element of (93, 1 2 ~ ), 1 b u t not of (23, 2lU1). Example 2.12. Let X be the real line, 93 the collection of the sets { n , l / n ) (n = 1, 2, . . . ). Then 2IU1and 2 ~ agree / on 93, and 23 is a closed (u ith either topology on 2X) collection of compact sets whose union is not closed. I t is apparent from the above examples that, except for Proposition 2.7, our various topologies are, in general, incomparable on 2X and its subcollections. Nevertheless, some simple and useful relations hcld betn een the finite and uniform topologies, and the next section is devoted t o their study. 3. Relations between the various topologies on 2X. For use i r i some of the proofs of this section, we introduce the follou ing notation for uniform spaces: Notatzon 3.1. Let [X, U ] be a uniform space, with index set A. Then, for all a E A : 3.1.1. Z(a) = {PEA]x, y, z E X : t : x E lTD(z), y E l.;l(z) t x € V,(z)) (see [16, p. 71). 3.1.2. If E C X , then V,(E)=UZEcV,(x). \\re now state and prove the fundamental lemma of this section. / LELIXIA 3.2. Suppose [X, 0-1 is a uniform space, and let E E 2 " . Then (see below) 3.2.1 e 3.2.2, and 3.2.3 e 3.2.4. 3.2.1. Any 2 ~ -neighborhood 1 of E contains a 2U-neighborhood of E . 3.2.2. E is totally bounded(g). 3.2.3. Any 21U-neighborhoodof E contains a 2 ~-neighborhood / of E . 3.2.4. If F E 2 = is disjoint from E , then there exists a n a such that V , ( E ) is disjoint from F. / / Proof. 3.2.1 t 3.2.2. Let a E A , and pick P E B ( a ) . Then there exists a of E which is contained in a a ( E ) . Let x , E 5, neighborhood (GI, . . . , nE ( i = 1, . . . , n). \Ve assert t h a t the sets Ve(sl), . . . , V,(x,) cover E . For let x E E . Then for some i, say i =k , U,C lTo(s). Hence x1.E VB(x),and hence x E V,(xk). 3.2.2 t 3.2.1. Let %,(E) be a / 2 ~-neighborhood / of E . Pick ,OEZ(a), y E Z ( P ) , and GEZ(y). Then there exist points xl, . . . , s, in E such t h a t the sets I/a(xl), . . . , Va(x,) cover E. Hence for each x E E , there exists a n i, saJi = k, such t h a t x E Va(x,). Hence x h E l.',(x), and hence JT8(xk)C V,(x). There(9) uniform space X is totally bounded if for every index w there exists a finite suhset (xi,. . . , x,] of X such that X=u:=, V,(xi). A\ 160 ERKEST MICHAEL [July fore, finally, ( V ~ ( x l ) ., . . , V~(X,))is a 21UI-neighborhood of E which is contained in 8,(E). 3.2.3 t 3.2.4. Let F E 2 X be disjoint from E. Then (F') is a 2IUI-neighborhood of E, and hence there e ~ i s t san a E A such that %,(E)C(F1). But then V,(E) C F', as asserted. 3.2.4 t 3.2.3. Let (Ul, . . . , U,) be a 2l"I-neighborhood of E. Let a E A be such that V a ( E ) r \ ( U ~ [email protected]'. l Let x , E U , n E ( i = 1, . . . , n), and pick p such that Vo(x,)C L7, ( i = 1, . . . , n). Let y z m a x (p, a). Then %,(E) C(L-1, . . , U,). Q.E.D. We now draw several conclusions from Lemma 3.2. T h e first of these, Theorem 3.3, has the widest application ; by virtue of it, any theorem proved about C ( X ) using either the finite or the uniform topology will automatically be valid in both topologies. In Theorem 3.4, we consider a special uniform structure on a normal space, and our conclusion is much stronger than that of Theorem 3.3; we use Theorem 3.4 in the proof of Theorem 4.9.5. THEOREM 3.3. If ew. [x, U ] is a uniform space, then 2IUI and 1 2 ~ agree / on Proof. This follo\~-sfrom Lemma 3.2 and the fact that if E C X is compact, then E satisfies conditions 3.2.2 and 3.2.4 [I, p. 111, Proposition 21. Q.E.D. THEOREM 3.4. If X is normal, and if U is the uniform structure induced on X by the Stone-tech compacti$cation, then 2Iv1 and / 2"/ agree on 2X. Proof. \\re will prove this theorem by showing that every E E 2 X satisfies conditions 3.2.2 and 3.2.4. Let X be the Stone-tech compactification of X. Since [X, U] is precompact, it is totally bounded [ I , p. 111, Theorem 41, and so is every E E 2 X . Let E and F be disjoint, closed subsets of X , and let E and F be their closures in X. Then we assert that E n F = @': For suppose that there exists an x o € E n F . By normality, there exists an f : X + [o, 11 such that f ( E ) =O and f ( F ) = 1. Now by the Stone-Cech theorem, f can be extended t o a continuous function f on X; but no such f could be continuous a t xo. This proves the assertion. Now since i? and are disjoint compact subsets of a uniform space, there exists an index a such that FA V,(E) = [ I , p. 111, Proposition 21. Hence FA V,(E) = @. Q.E.D. In contrast to Theorem 3.4, we finally have the following proposition. PROPOSITION 3.5. If the uniform structure of [X, U] is metrizable, then 21UI ) is compact. agrees with 1 2 ~ 1on 2Xif and only if (X, I UI Proof. If X is compact, then 21ul agrees with 1 2 ~ 1on 2X by Theorem 3.3. Suppose, conversely, that 2IUI and 1 2 ~ 1agree on 2X. Since 1 2 ~ 1is metrizable by 4.1.1, the same is true of 2Iv1, whence (X, ) is compact by Theorem 4.6. Q.E.D. This completes our study of the relations between the finite and the uni- I UI 19511 TOPOLOGIES ON SPACES OF SUBSE'TS 161 form topologies. We co~lcludethis section with the following proposition, which was already announced in the introduction, and which follo\vs from the definitions [3, p. 29, Ex. 71. PROPOSITION 3.6. Let X be a bounded metric space, and let U be the u n i f o r m strz~ctz~re which the metric induces o n X . T h e n the l ~ n i f o r mstrl~cturewhich the Hausdorff metric induces o n 2X i s identical with 2U. 4. Relationships between properties of X and 2X (or C ( X ) ) . \Tie begin by collecting some results about uniform spaces. PROPOSITION 4.1. Let [X, U] be a u n i f o r m space. 4.1.1. [X, U] i s metrizable F! [2X, 2U] i s metrizable. 4.1.2. [X, U] i s totally bounded F! [2X, 2U] i s totally bounded. 4.1.3. S u p p o s e that [X, U] i s nzetrizable. T h e n [x,U] i s complete F! [2X, 2U] i s complete. 4.1.4. There exist (nonmetrizable!) complete [x,U] for which [2X, 2U] i s not complete. Proof. T o prove the nontrivial half of 4.1.1, we re-metrize [X, U] by p' = p / ( l +p), and then apply Proposition 3.6. For the nontrivial parts of 4.1.2 and 4.1.3, see [I, p. 114, Ex. 51, and [12, p. 1981 or [3, p. 29, Ex. 71 respectively. Finally, 4.1.4 follows from the last part of Proposition 2.6 and the footnote on p. 37 of [2]. (This proof for 4.1.4 was suggested to the author by J. Dieudonnh.) Q.E.D. Throughout the remainder of this section, 2X will be assumed to carry the finite topology. T h e hard part of the following fundamental theorem was first proved by Vietoris [14]. THEOREM 4.2. X i s compact 2X i s compact. Proof. If X is compact, then 2X is proved compact by using Alexander's lemma, which asserts that a space is compact provided every covering by sub-basic open sets has a finite subcovering; see [7, Theorem 15(3) 1. Suppose, conversely, that 2X is compact, and let ( U,} be a covering of X by open sets. Then ( ( X , U,) is a covering of 2X by open collections, whence there exists a finite subcovering ( ( X , Ul), . . . , (X, U,)}. But then ( U1, . . , U,} is a finite subcovering for X. Hence X is compact. Q.E.D. Observe that in the second part of the above proof, we actually proved that if 2 X > G > J1(X), and if G is compact, then X is compact. Observe also that this part of the proof is unnecessary if X is assumed t o be Hausdorff, since a closed subset of a compact space is compact (see 2.4.2); a similar remark applies to 4.4.1 below. Observe that a uniform space [x, U] is compact if and only if [2X, 2U] is compact. This follows from Theorems 4.2 and 3.3; it can also be proved 162 ERNEST LIICHAEL [July directly [ I , p. 114, Ex. 61. The following lemma and its consequences deal with local compactness. LEMMA4.3. 4.3.1. I f X i s locally compact and A E C ( X ) , then there exists a n open U>A such that T7 i s compact. 4.3.2. A E 2 X has a compact neighborhood in 2X i f and only if A i s compact. Proof. 4.3.1. This is trivial. 4.3.2. (a) T o prove the "if" assertion, let U be the open set whose existence is asserted in 4.1.1. Then (D) = Cl ((U)) is a compact neighborhood of A (see Theorem 4.2). (b) We shall outline the proof of the converse. Let B C X be not compact, and let U = (U1, . . . , U,) be a neighborhood of B. Let U = Uy=,Ui; then is not compact. T o complete the proof we must show that, if were compact, then T7 would be compact. This demonstration is elementary but dreary, and is omitted. Q.E.D. u PROPOSITION 4.4. 4.4.1. X i s locally compact e 2X i s locally compact. 4.4.2. I f X i s locally compact, then C(X) i s open in 2X. Proof. 4.4.1. If X is locally compact, then 2X is locally compact by the "if" part of 4.3.2. Suppose, conversely, that 2X is locally compact, and let x E X . By Lemma 2.3, there exists a neighborhood V of x such t h a t C1 ((V) n C ( X ) ) (closure relative to C(X)) is compact. Then J1(V)cC1 ( ( V ) n C ( X ) ) c(2V, 2~12V)= (27, 2TIV),and hence 7 is compact by the comment following Theorem 4.2. 4.4.2. This follows immediately from 4.3.1 and 2.2.2. Q.E.D. We shall study next the question of countability and metrizability. L17e call a topological space separable if it has a countable dense subset: we call it second (Jirst) countable if there is a countable basis for the open sets (for the neighborhoods of each point). PROPOSITION 4.5. 4.5.1. X i s separable F! 2X i s separable. C(X) i s second countable. 4.5.2. X i s second countable 4.5.3. X i s $rst countable C(X) i s $rst countable. Proof. The implications pointing to the left are all obvious. Those pointing t o the right all follow easily from the fact that the family of finite subclasses of a countable class is countable (and from 2.4.1 in the case of 4.5.1). Q.E.D. THEOREM 4.6(7). I f (2x, 2T) i s either metric or second countable, then ( X , T ) i s compact. Proof. Suppose X were not compact. Then, in either case, it is not count- 163 TOPOLOGIES ON SPACES OF SUBSETS 19511 ably compact, and hence there exists a countably infinite A C X with no limit points. Hence A is closed in X , 2 A C 2 X , and A is discrete in the relative A . Now it is easy to see that 2TIA is identical with the relative topology topology 2 ~ 2A 1 which 2T induces on 2 A ; for the rest of the proof, we shall assume that 2A carries this topology. Now 2A is separable (see 4.5.1), so in either case 2A is second countable. We complete the proof by showing that this is impossible. For let be a basis for 2A, and for each E C A , pick a basis element UE such that E E U E C ( E ) . Now if El Z E z , say EZ-EI #@, then EZEUE, and E Z e U e , ; hence UE1#UE2. I t follo~vsthat ( cannot be countable, for A has more than countably many subsets. Q.E.D. The next proposition and its corollary deal with the extension of functions from X to 2 I . Proposition 4.7 is used in the proof of 4.9.4 and 4.9.9. We omit the proofs, which consist of straightforward verification. TI {u,} u,I PROPOSITION 4.7(7). Let X be a space, R the exteltded real line (that i s , R with a and added, in the order topology [2, p. 8 6 ] ) ,and f : X 4 8. Define f+: 2X + W and f-: 2 I -+ 8 by f + ( E ) sup,^^ f ( x ) , f - ( E ) = i n f Z E ~ f (x). T h e n f+ and f- have the same bounds (finite or injinite) a s f . I f T i s a topology (resp. U a u n i f o r m structure) o n X , and i f f i s continuous (resp. has its range i n R and i s uniformly continuous) with T (resp. U ) o n X , then f+ and f- are continuous (resp. u n i f o r m l y continuous, provided their range i s contained in R). + COROLLARY 4.8. Let ( X , T )be a topological space ( [ x , U ]a u n i f o r m space), conszdered a s a subspace of ( 2 X , 2 T ) (resp. [ 2 I , 2 U ] ) . T h e n a n y continuous ( u n i formly continuous) real-valued function f o n X can be extended to a continuous (uniformly continuous) real-valued function of C ( X ) ; i f f i s Jinitely bounded above or below, then f can be so extended to all of 2 I . Our next theorem deals with separation properties of 2X and C ( X ) . THEOREM 4 .9. 4.9.1. 2 I i s always To. 4.9.2. X i s T1 t 2 I i s T I . ( T h e converse i s false.) 4.9.3. X i s regular F! 2X i s Hausdorff. 4.9.4. X i s completely regular F! 2X i s a Stone space(lO). 4.9.5. T h e following are equivalent: (a) X i s normal; (b) 2X i s completely regular; (c) 2 X i s regular. 4.9.6. X i s compact Hausdorff F! 2 X i s compact Hausdorff. 4.9.7. T h e following are equivalent: (a) X i s compact and metrizable; (b) X i s second countable, compact, and Hausdorff; ' (lo) A topological space X is called a Stone space if, whenever xa and X I are distinct points in X, there exists a continuous real-valued function on X such that f(xo) =0, f(x1) = 1. 164 ERNEST RIICHAEL (c) 2X is compact and metrizable; (d) 2X is second countable, compact, and Hausdorff; (e) 2X is metrizable; (f) 2X is second countable and Hausdorff. 4.9.8. X is Hausdorff T? C(X) is Hausdorff. 4.9.9. X is a Stone space F! C(X) is a Stone space(lO). 4.9.10. X is regular F! C(X) is regular. 4.9.11. X is completely regular ",C X ) is completely regular. 4.9.12. X is compact (locally compact) Hausdorff F! C(X) is compact (locally comfiact) Hausdorff. 4.9.13. X is metrizable F! C(X) is metrizable. Proof. 4.9.1. If A , B E 2 X , B - A # @ , x E B - A , then B E ( X , A'), A B ( X , A'). 4.9.2. Given A , B E 2 X , B - A # @ , x E B - A . Then (a) B E ( X , A'), A B ( X , A'); (b) ~ ~ ( { x f ~' )B ,( { x f ' ) . As a counter example for the converse, consider a many-point space on which the only open sets are the whole space and the null set. 4.9.3. (a) Suppose X is regular. Given A , B E 2 X , B-A #@, x E B -A, let U, V be open sets such that A C U , x E V, and U n V = @ . Then A E ( U ) , B E ( X , V), and ( U ) n ( X , V)[email protected] (b) Suppose X is not regular. Then there exists an A E 2 X , and an x G A , such that A and x cannot be separated by open sets in X. But then it follows easily that A and A U { x ) cannot be separated by open collections in 2X, whence 2X is not Hausdorff. 4.9.4. (a) Suppose X is completely regular. Let A , B be distinct elements of 2X, and suppose that B - A #O, x E B -A. Let f be a continuous function from X t o the interval [0, 11 which is 0 a t x and 1 on A. Then, by Proposition = s u ~ , ~ E ~is( xcontinuous, ) and it is clearly 1 a t B and 4.7, the function f+(E) 0 at A. (b) Suppose 2X is a Stone space. Let A C X be closed, and let x E X -A. Let F be a continuous function from 2X to the interval [0, 11 which is 1 a t A and 0 a t A U { X ) . Let f : X -+ [0, 11 be defined by f(x) = F ( A U ( X ) ) . Then f is clearly continuous, and furthermore f is 1 on A and 0 a t x. 4.9.5. a + b. This follows from Theorem 3.4. b + c. This is obvious. c + a. Suppose t h a t 2X is regular. Let A E 2 X , and let U be an open set containing A. Now (U) is an open neighborhood of A and, therefore, since 2X is regular, there - exists an open neighborhood (Vl, . . . , V,) of A , such that (81, . . . , V,)C(U) (see 2.3.2). Let V=U;"=, Vi. Then A C V , and 8 C U (see 2.3.1). 4.9.6. This follows immediately from 4.2 and 4.9.3 above. d is well known. T h a t a + c follows from 4.2, 4.9.7. T h a t a F! b and c 3.3, and 4.1.1, while b t d follows from 4.5.2, 4.2, and 4.9.3. T h a t c + e and 165 TOPOLOGIES ON SP.4CES O F SUBSETS 19511 d t f is obvious. Finally e t a follows from one half of Theorem 4.6, while f tb follows from the other half. In the remaining assertions of this theorem, the implications pointing to the left all follow from the fact that X is homeomorphic to a subcollection of C(X). I t remains to prove the implications pointing t o the right. 4.9.8. This goes like 4.9.3 t , remembering that, in a Hausdorff space, a point and a disjoint compact set can be separated by open sets. 4.9.9. The proof proceeds just like the proof of 4.9.4, if we remember that in a Stone space a point and a disjoint compact set can be separated by a continuous function. 4.9.10. Let A E C ( X ) ; let (UI, . . . , U,) be a neighborhood of A , and let U = U L l Ui. I t suffices t o show that we can find a neighborhood (Vl, . . , V,) of A whose closure is contained in (Ul, . . , U,). Now A C U , and since A is compact and X is regular, there exists an open V I A such that V C U. Let x i € U i n A ( i = 1, . . . , n). Since X is regular, there exists, for each i, a neighborhood Vi of x i such that Tic Ui (i = 1, . . . , n). But then (I.', Vl, . . . , V,) is a neighborhood of A , and (see 2.3.2) . C1 ((V, Vl, . . . , V,)) = - (P, 71,. . . , V,) C (Ul, . . . , un). I UI 4.9.11. Let U be a uniform structure on X such that = T. By Theorem 3.3, 2T is equivalent to 1 2 ~ 1on C(X), when (C(X), 2T) is completely regular. 4.9.12. If X is compact (locally compact) Hausdorff, then so is C(X) by 4.9.6 (by 4.4.1 and 4.9.8). If C(X) is compact (locally compact) Hausdorff, then so is X by 2.4.2. = T. 4.9.13. Let U be a metrizable uniform structure on X such that Then, again, 2 T = 1 2 ~ 1on C(X). The assertion now follows from 4.1.1. Q.E.D. Note. 4.9.5 a t c could be proved directly without using Theorem 3.4; see the proof of 4.9.10. T h e remainder of this section will deal with connectivity. A feature of 2X with the Hausdorff metric (and with X compact) is that it has stronger connectivity properties than X itself (see [ l l ] ,especially the introduction). We shall not go into analogous results in this paper. We shall also not study the properties of the collection of connected, respectively locally connected, subsets of X (see [ s ] , [ l l ] , and [IS]). When studying connectivity, it is more convenient to look a t all of d ( X ) than a t 2I. There is an obvious extension of the finite topology to d ( X ) (see Definition 5.1 for definition and notation), and for the remainder of this section, we shall assume that d ( X ) carries this topology. I UI THEOREM4.10. Let J ( X ) c G c d ( x ) . If one of the spaces X , J,(X) (n = 1, 2, . . ), or G is connected, then all of them are connected. . Proof. (a) Suppose that X is connected. Then, by 2.4.3, J,(X) is connected 166 ERNEST I\/IICHAEL [July for all n. If El FE J(X), then they both belong to some (connected!) J,(X); hence J(X) is connected. But J(X) c G c d ( X ) = C1 (J(X)) (see 2.4.1), and therefore G is also connected. (b) If J,(X) or G is connected, then, by 2.8, X is also connected. PROPOSITION 4.11. Let A1, . . .-, A, be connected subsets of X. I f J(X) n ( ~. ~ . . , A,)+cGc(&, . . . , A,)+, t h e n G i s connected. Proof. By Theorem 4.10, J(Ai) is connected ( i = 1, . . . , n ) ,and therefore But J ( X ) n ( A 1 , . . . ,An)+isthe J(A1) X . . X J(An) is connected in [&(x) image of J(A1) X . . X J(A,) under the (continuous!) map discussed in 5.8.1, and therefore J ( X ) n ( A l , . . . , An)+ is also connected. Finally G is contained between a connected collection and its closure (see 2.3.2), and therefore G is also connected. Q.E.D. In. THEOREM 4.12. Let J ( X ) C G C C ( X ) . T h e n X i s locally connected i f a n d only i f G i s locally connected. Proof. (a) Suppose that X is locally connected. If E E C ( X ) , and if U is a neighborhood of E in d ( X ) , then we can find connected, open sets Ul, . . . , U, such that E E ( U 1 , . . , U,)CU. I t now follows easily from Proposition 4.1 1 that G is also locally connected. (b) Suppose, conversely, that G is locally connected. Let x E X , and let U be a neighborhood of x. Then there exists a connected neighborhood 2.3 of { x } in G , such that 8 C ( U ) . Therefore V = U A ~ is~ aA neighborhood of x, V C U and 8 is connected (by 2.8) since { x }€8 and { x ) is connected. Q.E.D. Theorem 4.10 is false with the uniform topology on d ( X ) . (If X is an unbounded metric space, with uniform structure U induced by the metric, then the collection of bounded sets is open and closed in ( d ( X ) , 2 ~).)/ Theorem 4.12 becomes false if C(X) is replaced by 2X or d ( X ) . (If X is the real line, I the set of integers, and U = (UnEISIIlo(n)),then there is no connected neighborhood of A contained in U.) In the following proposition we collect some of the remaining facts about connectedness in 2X and C(X). They are all easily verified. (We call a space totally disconnected if any two distinct points can be separated by an open and closed set.) / PROPOSITION 4.13 (7). 4.13.1. X i s zero-dimensional C(X) i s zero-dimensional. 4.13.2. X i s totally disconnected F! e ( X ) i s totally disconnected. 4.13.3. X i s discrete C(X) i s discrete. 4.13.4. X has n o isolated points F! 2X h a s n o isolated points. 4.13.5. T h e collection of connected elements of 2X i s closed in 2 I . 5. Functions from or to hyperspaces. In this section, it will be useful to 1951j 167 TOPOLOGIES ON SPACES OF SUBSETS extend our topologies and uniform structures to & ( X ) (see Notation 1.4). We make the following definition. DEFINITION5.1. 5.1.1. If ( U , ) L E Iis a collection of subsets of a topological space X I then ( U L ) k I denotes ( E C ~ ( X ) I E C U , ~ ~ UE, A; U , # @ for all L E I ) .F or a uniform space, 23,+(E) denotes (V,(x))ZEE. 5.1.2. The finite and uniform topologies on & ( X ) are defined just as in Definitions 1.6 and 1.7, except that (-) is now replaced by (-)+. 5.1.3. The map q 5 : d ( X ) + 2 X is defined by q5(A) In the following corollary, A stands for the relative topology induced on A by the topology T o n X > A . Similarly for uniform structures. TI =x. COROLLARY 5.2. Let X be a topological ( u n i f o r m ) space. T h e n : 5.2.1. ( U , ) , E r = 2 " A ( U , ) k I . 5.2.2. T h e j i n i t e ( u n i f o r m ) toeology o n 2" i s the relative toeology induced o n 2 X b y the finite ( u n i f o r m ) topology o n d ( X ) . 5.2.3. I f ( X , T ) i s a topological seace a n d i f A C X , then (&(A), 2TIA) = (&(A), 2 ~ 2A). 1 5.2.3'. I f [ X I U ] i s a n u n i f o r m space, a n d if A C X , then [&(A), 2UIA] = [&(A), 2 ~2A]. / 5.2.4. I f A C X i s dense, then 2A i s dense in & ( X ) in the jinite or u n i f o r m topology. A fundamental relation between & ( X ) and 2 X is established by the following theorem. THEOREM 5.3. 5.3.1. I f ( X I T ) i s normal, then q5 i s a retract.ion. 5.3.1'. I f [x,U ]i s u n i f o r m , then q5 i s a u n i f o r m retraction. Proof. In both cases, 2 X evidently remains pointwise fixed. I t remains to show the (uniform) continuity of 6. 5.3.1. Let E E d ( X ) , and let U = ( U l , , U,) be a neighborhood of E. Pick Vo such that EC Vo and VoCU:="=,j, and let 23 = ( VoA U1, , V O AU,)+. Then 23 is a neighborhood of E in & ( X ) , and q5(23)C U . 5.3.1'. To index a for 2 X , assign any P E Z ( a ) for d ( X ) . Q.E.D. COROLLARY 5.4. I f ( X , T ) i s a topological space ( [ X I U]a u n i f o r m space), a n d A C X , then q51 2* i s 1-1 a n d continuous ( u n i f o r m l y continuous) w i t h the finite ( u n i f o r m ) fopology o n both 2 A a n d 2 X . \Ve nonr define some inlportant functions and derive their properties. DEFISITION5.5. 5.5.1. a : d ( d ( X ) ) -+ d ( X ) is defined by a ( % ) = UEEeE. 5.5.2. 0 : 2 X -+ d ( 2 " ) is defined by 0 ( A ) = { E E ~ X ~ E C=2". A) 5.5.3. 77:2X -+ ~ 4 ( 2 is~ defined ) by q ( A ) = ( ~ € EAA 2 ~ 1 #@I. 3.68 ERNEST MICHAEL [July All statements in Corollary 5.6 are immediate consequences of previous results; following each statement, we list the relevant references. COROLLARY 5.6. 5.6.1. If X i s regular, then U ( C ( 2 ~ T~) ), C 2 X 5.6.2. a ( C ( C ( X ) ,2 T ) ) C C ( X ) 5.6.3. 8 ( 2 ~ ) ~ 2 ( 2 ~ . 2 ~ ) 5.6.4. B(C(X))C C ( C ( X ) ,2 T ) 5.6.5. 7 ( 2 ~~ ) 2 ( ~ ~ * ~ ~ ) If [ X , U ] is a uniform seace, then 5.6.1'. U ( C ( ~1 2~ ~, 1 )C) 2 X 5.6.2'. de(c(x),2 Ul )) 5.6.3'. 0 ( 2 ~ ) ~ 2 ( 2 ~ ~ 1 2 ~ 1 ) (5.6.4 and 3.3), (2.2.2'). THEOREM 5.7. If [ X , U ] is a uniform @ace, and if all hyperspaces aepearing in DeJinition 5.5 carry the uniform topology, then 5.7.1. The functions a, 8 , and 7 are all uniformly continuous. If ( X , T ) i s a topological space, and i f all hyeerspaces in DeJinition 5.5 carry the Jinite topology, then: 5.7.2. a is continuous, 5.7.3. 81 C ( X ) i s continuous, 5.7.4. 7 is continuous i f X is compact. 1 cecx) Proof. 5.7.1. T h e essential fact in the proofs of this statement is that the index sets for X , c A ( X ) ,a n d c A ( d ( X ) )are all the same. For each function, we shall give a rule which assigns to each index for the range an index for the domain. T h e verification of this rule is straightforward for a, and a little more complicated for 8 and 7 . a : T o a for c A ( X ) assign a for d ( d ( X ) ) .Let 8 € a ( 8 )= B . d ( Let DE(V,(E))E+E4t, a ( b )=D. We must show that D E % ; ( B ) . (a) D C V , ( B ) : Let d E D . Then there exists an F E D such that d E F . But there exists an E E 8 such that FEfl32S,f(E). Therefore there exists an e E E C B such that d E V,(e). ( b ) D n V,(x) # @ for all x E B : The proof is similar to the one for (a), and we omit it. 8:T o a for 22X assign a P E Z ( a )for 2 X . Let A E 2 X ,8 ( A )=%. Let B E % p ( A ) , 8(B)= 8 . We must show that B E ( V a ( E ) ) E E a : (a) B n @ B ( E#) @ for all E E % :Let E E 8 . Then E C A . Let H = y E V B ( xfor ) some x E E ) . Then H C B , whence BcB = B , and therefore B E 8 . We shall show that W E % , ( E ) : ( 1 ) B n V , ( x ) #% for all x E E : Let x E E . Then there exists a Y E H C B such that y E V b ( x ) CV,(x). ( 2 ) BC V , ( E ) :Let ~ € Pick y E H such that z7E7 .V,.(y),where y E Z ( P ) . { y ~ ~ I TOPOLOGIES ON SPACES OF SUBSETS 19511 169 Next, pick x E E such that y E Vb(x).Then z E V,(x). (b) %CUEE%%,(E): The proof is similar to that for (a), and we omit it. 77 : T o a for 22Xassign /3 for 2X: The proof is similar to the one for 8 , and will be omitted. 5.7.2. Let % E d ( d ( X ) ) , a(%) = B , and let U = (Ul, . . . , Un)+ be a neighborhood of B. We must find a neighborhood % of 23 such that a(%)CU. ~ 1U ] ; then UOis open in 2X, by 2.2.2. Let U = U;=, U;, and let UO= ( ~ € 2 EC Let Ui = (Ui, U)+ ( i = 1, . . . , n). We now define % = (UO,U1, . . . , ITn)+. (a) '$1 is a neighborhood of 23: Let xi€ U i n B , and pick B i E % such t h a t x i E B i ( i = l , . . . , n). Then B i E ( U ; , X ) + ( i = l , . . . , n), and%CUo. (b) a(%)CU: Let XI€%,and let a(D) = D. (1) D C U: Let X E D . Then there exists an E E D such that x E E . But E E U , so EC U, whence x E U. (2) D n Ui#@ for all i: Pick E,CD such that E E U ; ( i = 1, . . . , n). Now pick y i E E i such that y i E Ui ( i = 1, . . . , n). Then y i E D A U ; ( i = l , . . . a). 5.7.3. If X is completely regular, then it admits a uniform structure, and our assertion follows from 5.7.1, 5.6.4, and 3.i. The assertion can be proved directly, without assuming complete regularity (but using the restriction to e ( X ) ) . The proof is somewhat messy, and is omitted. 5.7.4. If X is Hausdorff, then, as in 5.7.3, the assertion follows from 5.7.1, 4.2, and 3.3. Here again the assertion could be proved directly, without assuming Hausdorffness (but using the compactness of X ) . Q.E.D. Concerning the function a , we establish the following additional conclusions: c.Z(cr) 7 COROLLARY 5.8. Let X be a topological (uniform) space. Then 5.8.1. The map from [ d ( X ) 1%into d ( X ) which sends (El, . . . , E n ) into U:=' Ei is continuous (uniformly conkinuous) with the finite (uniform) topology on cA(X). 5.8.2. If,in 5.8.1, we map (E, F) into En F, bhe function is not continuous (uniformly continuous) (it is, however, upper semi-continuous ; see Appendix). The same applies to the function which we get if replaces V in the definition of a. 5.8.3. I n either topology, U(@)CCI ( a ( 8 ) ) for any % E & ( d ( X ) ) . Proof. 5.8.1. This follows from 5.7.2 (5.7.1) and 2.4.3. I t is also easy to prove directly. Q.E.D. \Ve now study functions between topological spaces. DEFINITION 5.9. Let f : X -+ Y be onto. We define 5.9.1. f * : d ( X ) -+d( Y) by f*(E) =f(E). 5.9.2. f-I*: Y -+ d ( X ) by f-l*(y) =f-l(y). 5.9.3. f-'**:cA(Y) + d ( X ) by f-l**(E) =f-l(E). 5.9.4. OCf)= (f-'(y))yEY=f-'*(~). 170 ERNEST MICHAEL [July Before stating the fundamental theorem on the functions defined above, we recall (make?) the following definition about uniform spaces which will be used in 5.10.2' below: Let f :X--t Y be a function between uniform spaces. Then f is uniformly open (resp. closed) if to every index a for X there corresponds an index a' for Y, such that for every x E X ( Y EY) we have f(V,(x)) 3 V,,(f(x)) (resp. f [ ( v a ( f - l ( ~ ) ) ) ' I ~ ( V a , ( ~ ) ) ' ) . THEOREM 5.10. Let X, Y be topological spaces, and let f :X+ Y be onto. Then' wzth the jinite topology: 5.10.1. f * is continuous (a homeomorphism) if and only iff is continuous (a homeomorphism). 5.10.2(8).f-l* is continuous if and only iff is open and closed. 5.10.3. f-I** is continuous if and only zf f-I* is continuous. Now let X , Y be uniform spaces. Then: 5.10.1'. f * is z~niformlycontinzroz~s(an isomorphzsm) $and only iff is uni- formly continuous (an isomorphism). 5.10.2'. f-l* is uniformly continuous if and only iff is uniformly open and uniformly closed. 5.10.3'. f-I** is uniformly continuous if and only if f-'* is uniformly continuous. If D(f) carries the factor topology, then 5.10.2". Iff is either open ov closed, then f-I* is continuous. Finally we have 5.10.4. Iff is continuous, open, and closed, then the jinite and the factor topology on D(f) are equivalent. Proof. Assertions 5.10.1 and 5.10.1' follow directly from the definitions. 5.10.2" is well known and trivial. The proof for 5.10.2' (5.10.3') is t h e same as for 5.10.2 (5.10.3). Finally, 5.10.4 follows immediately from 5.10.2 and 2.7. 5.10.2. (a) Suppose f is open and closed. Let y E Y, and let fP1*(y) = E. Let U = ( U l , . . . , U,)+ be a neighborhood of E, and let U = Uy=L=,U;. Let Vl= Uy=lf(U;) ; since f is open, Vl is open. Let V2 = Cf(U1))';then y E V2, and V2 is open since f is closed. Letting V= V l n V2, we see that f-'*(V)CU. (b) Suppose that f-I is continuous. If U C X is open, then f ( U ) = up1*)-'((X, U)+) which is open; so f is open. If A C X is closed, then f(A) = ((f-'*)-'((A1)+))' which is closed; so f is closed. 5.10.3. (a) If f-I** is continuous, so is fW1*=f-'**I JI(Y). (b) If f-l* is continuous, so is fwl** =ao(f-I*) *. Q.E.D. From Theorem 5.10 it follows that if X and Y are homeomorphic (uni- formly isomorphic) so are 2X and 2Y. T h e converse, however, is false; see [s]. In the special case where the domain X is compact, we have: PROPOSITIOX 5.11. Let f : X-+ Y be a continuous map from a compact Hausdorff space X onto a Hausdorff space Y. Then: TOPOLOGIES ON SPACES OF SUBSETS 19511 171 5.11.1. If D(f) is closed in 2X, then the finite topology on DCf) is equivalent to the factor topology. 5.11.2. f-l* is continuozcs if and only if D(f) is closed in 2X. Proof. 5.1 1.1. By Proposition 2.7, we need only show t h a t the finite topology is coarser than the factor topology. Suppose GCD(f) is closed in D(f) in the finite topology. Then (3 is compact, whence a ( E ) is closed (see 5.6.1); that is, G is closed in the factor topology. 5.11.2. If D(f) is closed in 2X, then f-l* is continuous by 5.11.1 and 5.10.2". If f-l* is continuous, then DCf) =f-l*(f(X)) is compact, and hence closed. From 5.10.2 and 5.11.2 we conclude that: COROLLARY 5.12. Iff:X-+ Y is a continuous map from a compact Hausdor$ space X into a I$ausdorff space Y, then f is open if and only if D(f) is closed in 2I. Before concluding this section, we shall define an interesting new concept and derive its principal properties. DEFINITION5.13. Let (X, T) be a topological space, and let 2 3 E d ( 2 I ) . 5.13.1. The saturate of 23 (written sat (23)) is A is a union of elements of % ) . 5.13.2. 8 is saturated if sat (23)=%. 4 s immediate consequences of this definition we have the following corollary. {AE~XI COROLLARY 5.14. If 23 E d ( 2 X ) , then: 5.14.1. sat (23) E d ( 2 X ) . 5.14.2. s a t (sat(%))= s a t (a). 5.14.3. a(sat (%))=a(%). 5.14.4. If 23 is saturated, and -f A E 2 I , then 23T\O(A) and %T\a(A) are both saturated. \Ve further have the following lemma. LEMMA5.15. If % E d ( 2 I ) , then, in the finite topology: is aJinite union of elements of 93) is dense in sat (23). 5.15.1. 5.15.2. If 23 is saturated, then Cl(a(%))E%(closure of 23 in 2I). 5.15.3. If 23 is saturated, then Cl(a(8)) =a(@. (AE~XIA Proof. 5.15.1. Let E E s a t (%), and let U=(U1, . . . , U,) be a neighborhood of E. Pick xi€ U i A E ; then there exists an EiEB such that E i C E and such that x i E E i ( i = 1, . . . , n)., by the definition of saturate. But now Uy=l E i E I t , proving the assertion. 5.15.2. Let Lt=(Ul, . . . , U,) be a neighborhood of Cl(a(23)). Pick xi€ UiAa(23); then there exists an EiE23 such that x i E E , ( i = 1, . . . , n). But then U:='=, EiEBT\Lt. 172 ERNEST MICHAEL [July 5.15.3. This follows from 5.15.2 and 5.8.3. Q.E.D. . From 5.15.3. we have the following corollary, which should be compared to 2.5.1 and 2.5.2 (or 5.6.1 and 5.6.2). . COROLLARY 5.16. Let (X, T) be a topological space. If 93 E 2 ( 2X T ) zs saturated, then a(%)E 2 X ; that is, a closed saturated union of closed sets is closed. s 2 I t is not true, in general, that the saturate of a closed collection is closed. For compact collections, however, we have the following proposition. PROPOSITION 5.17. If X is a regular (uniform) space, and if ~ 4 ( 2 carries ~ ) the finite (uniform) topology, then: 5.17.1. If B E C ( 2 X ) ,then sat (%)EC(2X). 5.17.2. The function sat :C(2x)-+C(2x) is continuous (uniformly continuo~~~). Proof. By 5.6.4 (5.6.4'), 6(C(2X))CC(C(2X));by 5.7.2 and 5.10.1 (5.7.1 and 5.10.11),the fact that a continuous image of a compact space is compact, and by 5.6.1 (5.6.11),we have C T * ( C ( C ( ~ ~ ) ) ) C CFurthermore, (~~). 6 is continuous (uniformly continuous) by 5.7.3 (5.7.I ) , and a * is continuous (uniformly continuous) by 5.7.2 and 5.10.1 (5.7.1 and 5.10.1'). Hence the composite function a* 0 6 sends e(2X) into e(2X)and is continuous (uniformly continuous). T o complete the proof, we shall show that if % E e ( 2 X ) , then sat (8) =a*O(%). Now a*O(%) = E is the union of a compact subcollection of % ) ; since each element of a*O(%) is closed (see above), we have a*O(B)Csat (a). Conversely, if E E s a t (a),let T2 = { FE%[ FCE). Then E = a ( D ) and DEO(%); hence EEa*O(%), and hence sat (B)Ca*O(B). Q.E.D. This completes our study of the saturate. We conclude this section by stating, for later reference, the following lemma about general topology. {EE~XI LEMMA5.18. Iff:X-+ Y is a closed function from the regular (topological) space X onto the topological space Y, and if f-'(y) is closed (compact) for all y E Y, then f-'(C) is closed (compa.ct) for all compact C C Y. Note(8). If we assume that f is open as well as closed, then the lemma follows from 5.10.2 and 2.5.1 (2.5.2). 6. Multi-valued functions. Suppose that Y and X are topological spaces, and that g is a multi-valued function from Y into X . I t is then possible to regard g as a single-valued function from Y into d ( X ) , which is what we shall do from now on. Once d ( X ) has been topologized, g becomes a function between topological spaces, and it then makes sense to talk about its continuity and other topological properties. In this section we shall suppose that d ( X ) carries the finite topology, and that, furthermore, g maps Y into 2X (that is, that g, considered as a multi-valued function, maps each point of Y into a closed subset of X ) . We now ask the following question. 19511 TOPOLOGIES ON SPACES OF SUBSETS 173 Question 6.1. When is it possible to find a continuousg': Y --t X , such that gl(y) Eg(y) for all Y EY? A sufficient condition that this be possible is that 6.2. Both the follou~inghold: 6.2.1. g is continuous. 6.2.2. There exists a selection (see Definition 1.8) from 2X to X. The problem is thus reduced to two simpler ones; the second of which is concerned only with the space X , and has nothing to do with the space Y or the function g. Most of $7 will be devoted to determining what spaces satisfy 6.2.2. Now suppose that g has the special form g =f-I*, where f : X -+ Y is continuous. In this case, we call g' a continuous inverse for f , and we can improve the criteria of 6.2 as follows: 6.2'. Both the following hold: 6.2l.1. f is open and closed (see 6.1.1 and 5.10.2). 6.2l.2. For an arbitrary disjoint covering of X by closed sets, there exists a selection from to X . In $8 we shall give a few preliminary results about spaces which satisfy 6.2l.2. We shall see there that there are spaces which satisfy 6.2l.2 but not 6.2.2. Finally, suppose that, in addition, X is compact Hausdorff. If now f is a continuous function from X onto a Hausdorff space Y, then any continuous inverse g' off is a homeomorphism of Y into X , and g' 0 f is a retraction of X onto a subset homeomorphic to Y. We also have the following necessary and sufficient condition. , PROPOSITION 6.3. Let X be a compact Hausdorff space. Then the following are equivalent : 6.3.1. Every continuous open function from X to a Hausdorff space Y has a continuous inverse. 6.3.2. For every closed disjoint covering 2) of X by closed sets, there exists a selection from 2) to X . Proof. 6.3.1 + 6.3.2. Let 2) be a closed disjoint covering of X by closed sets. Let i : X -+ %'be the cononical inclusion map; by 5.11.1 it is continuous, and by 5.12 it is open. Now let g' be a continuous inverse for i; then g' is a selection from 2) to X. 6.3.2 + 6.3.1. Let f : X -+ Y be continuous and open. Then D(f) is closed, by 5.12, and f-I* is continuous, by 5.10.2 (or 5.1 1.2). Let h:!D(f) --t X be a selection. Then h 0 f-I* is a continuous inverse off. Q.E.D. 7. Selection on 21. Throughout this section, we assume that ( X , T) is a Hausdorff space, and that 2X carries the finite topology. We first prove some preliminary lemmas, from which our main results (in particular Theorem 1.9) will follow as corollaries. 174 ERNEST MICHAEL [Jul~ Notation. If < is an order relation on X , then I * ( x ) = ( t ~ ~< xl ) ;t I * ( x )= { t € ~ j t > x ] . DE:FINITION 7.1. If f : Js(X)+X is a selection, then x < y means x # y and f ( { x , y } ) =x(l1). Our first lemma draws conclusions from the existence of a selection on J z ( X ); it will soon be evident why JZ(X) plays a special part in our investigation. LER then : 7.2.1. 7.2.2. 7.2.3. 7.2.4. ~ 7.2. A If X is connected, and if there exists a selection f : J2(X) + X , For all x E X , I* ( x ) and I * ( x ) are open i n X . < is a proper linear order on X . The order topology on X is coarser than T . There exists exactly one other selection g : J 2 ( X )--t X , vzamely Proof. 7.2.1. I* ( x ): Let t < x . Then f ( { t , x } ) = t. Let U be a neighborhood of t such that x E U. Then, since f is continuous, there exists a neighborhood 1% of t and a neighborhood W Zof x such that f ( F ) E U for all F E ( W 1 , 1%). ~ e n c te' ~ W 1 - t .f ( { t l , x } ) € ~f (t( t t , x } ) = t l + t l < x . I * ( x ) . Let t >x. Then f ( { t , x ) ) =x. Let U , V be open in X such that x E U , tE V , UT\ V = @ . By continuity of f , there exists a neighborhood 1471 of x , and a neighborhood W z of y such that FETB = ( W l , W z )+f ( F ) E U. Let W = W Z A V , and let TBt=(1K, W ) . Then TBICTB, and therefore FETB' +f ( F ) E U . Hence t l € J V t f ( { x ,t t } ) € ~ +f ( { x , t l } ) = x (since t 1 E V 1so t l a U ) t t' > x. 7.2.2. We need only show that < is transitive. But this follows from 7.2.1 and [6, ( 2 . 1 ) ] . 7.2.3. This follows immediately from 7.2.1. 7.2.4. (a) g is a selection: If x, y E X , write x<<y for x # y , g ( { x , ] ) =x. y < x , 7.2.2 and 7.2.3 imply that << is a proper linear order on Since x<<y X such that the order (<<) topology is coarser than T . Applying Lemma 7.5.1, we find that g is a selection. (b) f and g are the only selections: Let h be any selection on J Z ( X ) . Then h generates an order on X such that the order topology is coarser than T (by 7.2.2 and 7.2.3). We now apply [6, Theorem 21; this tells us that the order generated by h is the one generated by f or its inverse. Hence h=f or h=g. Q.E.D. LEMMA7.3. Let X be connected. Then: 7.3.1. I f , for some$xed n , there exists a selection f : J , ( X ) + X , then f ( E ) (I1) Strictly speaking, we should write ever, omit this precaution. <,. Where no confusion can occur, we shall, how- 175 TOPOLOGIES ON SPACES OF SUBSETS 19.511 is the jirst (<) element of E for every EE J,(X). 7.3.2. If J(X) C E I C ~and ~ , if there exists a selection f : G --+ X , then f ( E ) is the jirst ( < ) element of E Jor every E EE. Proof. 7.3.1. Let BEJ,(X), x, y E B , and x < y . Let @ = ( E EJ,(x) J ~ (u E {x, y = y } . We shall show that both @ and J,(X) - Q are open. Since J,(X) is connected (by Theorem 4.10), and since {x, y ) E Jn(X) [email protected], it will follonr that @ is empty, and hence that f(B) #y. But this is the assertion of 7.3.1. (a) J,(X)[email protected] open: Let EEJ,(X)[email protected] Then f ( E U { x , y ) ) = z # y . Let U be a neighborhood of z such that y e U. Then, since f is continuous, there exists a neighborhood 9?= (Nl, . . . , Nk) of EU {x, y ) such that FE J,(X) 17% tf (F) E U. Let 92 be a neighborhood of E generated by those Ni which are neighborhoods of some element of E. Since the only Ni among the generators of % which are not among the generators of 9X are neighborhoods of x or y, wehave F E J , ( X ) A % I I t F U ( x , y ) E % t f ( F U ( x , y ) ) E U t f ( F U { x , y ) ) #y t F E J n ( X ) - @ . (b) @ is open: Let E E @ . Then f ( E U ( x , y 1) =y. Let U, V be open in X such that y E U, [(EU {x, y - ( y ) ]CV, and such that UI7 V= @. Then, by continuity of f , there exists a neighborhood % = ( N I , . . . , Nk) of E U ( x ,y ) such that F E 9 ? AJn(X) t f ( F ) E U. Let 9X = (MI, . . . 9 Mj) be the neighborhood of E which is derived from % as in (a) above, and = (MIA V, . . . , M,A V). Then FE J,(X)A9X1 t FU {x, y } E % let t f ( F U ( x , y ] ) E U t f ( F U { x , y } ) = y (since the only point of F U ( x , y ) , which lies in U, is y) t [email protected] 7.3.2. Let E E G , f ( E ) =y. Suppose that the assertion of 7.3.2 is false; then there exists an x E E such that x < y . Then I*(x) is a neighborhood of y, so there exists a neighborhood '2 of E such that F E G n % t f ( F ) E I * ( x ) . Now let F be a$nite set in G A R (such an F exists by Proposition 2.4.1) such that x E F . Then f ( F ) = z E I * ( x ) , z#x, so x <z. Let FE J,(X); then f J,(X) is a selection, and hence, by 7.3.1, x > z . We therefore have a contradiction. Q.E.D. The assertions of the next lemma follow directly from Lemmas 7.3 and 7.2, if we assume that X is connected. T o prove it under our weaker assumption, we use the fact t h a t any set can be well-ordered. 1) 1) 1 LEMMA7.4. Suppose that all the connected components of X are open. Then: 7.4.1. If there exists a selection f :J2(X) --+ X , then there exists a linear order on X such that the order topology is coarser than T. 7.4.2. If there exists a selection f : 2X -+ X , then there exists a linear order on X such that the order topology is coarser than T, and such that every closed (T) set has a $rst element. Proof. Well order the components of X . For "the component which contains x" we shall write "C(x)". 176 ERNEST MICHAEL [July 7.4.1. I t is evident that, for everp component C, f is a selection when restricted t o J2(C) = ( E EJ ~ ( x ) E C C } . Define an ordering on X a s follows: / s < y means m, if s and y are in the same component, C(n:) < C(y), if n: and y are in different components. This ordering is certainly linear, and furthermore, for everp x E X , I,(%) = f t E C(s) / t < s} r\ U C<C I*(%) = { t C, which is open (see 7.2.1), ( Z) E C(s) I t < s } n U C>C C, which is open (see 7.2.1). (2) 7.4.2. Evidently, for every component C, f is a selection on 2C = { E € 2 ~ 1EC C ) . W e define our order as in 7.4.1 ; i t remains t o show t h a t every closed ( T ) set has a first element. Let A be closed, and let C be the first component of X containing a point of A . Then A n C is a nonempty closed subset of C, and therefore contains a first element y by 7.3.2. B u t this y is evidently the first element of A , xirhich proves our assertion. Q.E.D. Our final lemma shows that, under certain conditions, there actuallqexists a selection from a subcollection of 2X t o X. Its proof is quite independent of the previous lemmas of this section, xirhich deal with t h e converse problem. L ~ h l h 7.5.1. l~ Suppose that there exists a linear order on X such that the order topology is coarser than T , and let X ( X ) = f E € 2 ~ 1 For e-dery F E 2 X , E n F is empty or has a first element). 7.5.1. Then f : X ( X ) -+ X , defined by f ( E ) =first element of E, is a selection. 7.5.2. Then C ( X , T ) C X ( X ) . Proof. 7.5.1. Let E E X ( X ) , and let f ( E ) = x . Let U be a n open ( T ) set containing x ; we must find a neighborhood % of E such t h a t F E % n X ( X ) t.f ( F ) E L'. Now if EC U, then 9? = { U ) is such a neighborhood, and we are through. Suppose, therefore, t h a t E C U , and let y be the first element of En U'. W e now consider two cases: (1) There is no z E X such t h a t x < z < y . Let N l = ( t l t < y ) n ~ ,N2 = f t l t > x ) . Then Nl and N2 are open ( T ) ; X E I V ~y, E N 2 ; and E C N l U N 2 . Hence 9? = (Nl, N2) is a neighborhood of E . B u t t l E Nl, t 2 E N 2t tl < z , t22z t tl<t2; so F E % n X ( X ) t.f ( F ) E N l c U . (2) There exists a z E X such t h a t x < z < y : Let N1 = U, N2 = ( t t < z } n U, N3= ( t l t > z ) . T h e n N1, N2, and N 3 are open ( T ) ; x E N l , x E N 2 , and y E N 3 . Also E E N l U N 2 U N 3 ; for if z E E , then z E I: by definition of y and z. Hence % = (Nl, N2, N3) is a neighborhood of E. Now tzEN2, t3E N 3 t.t2 < t3; and therefore F E % n X ( X ) t f ( F ) E N l U N 2 C U . 7.5.2. Let E E(S(X). Then EAF is compact ( T ) for all F E 2 X , and hence is also compact in the order topology, xirhence it has a first point. Hence 1 19511 TOPOLOGIES ON SPACES O F SLTBSETS 177 C ( X ) < X ( X ) . Q.E.D. \Tie are noxi7 ready to prove the principal results of this section. The proofs will consist of references to the lemmas from xirhich they folloxir. Proof of Theorem 1.9. 1.9.1 t 1.9.2: (7.4.2). 1.9.2 t 1.9.1: (7.5.1)(12). 1.9.3 +- 1.9.4: (7.4.1). 1.9.4 t 1.9.3: (7.5.1 and 7.5.2)(12) Q.E.D. The next proposition deals with the possibility of extending a selection from J 2 ( X ) to X . PROPOSITION 7.6. I f X i s connected, and i f there exists a selection f : J 2 ( X ) -+ X , thelz: 7.6.1. There exists a n extension o f f to C ( X ) . 7.6.2. I f euery E E 2 ' has a f i r s t ( < r ) element, then there exists a n extension o f f to 2 x . 7.6.3. I f J ( X ) C B C 2 X , then a n y extension to E i s unique. 7.6.4. For all n, a n y extension to J , ( X ) i s unique. Proof. 7.6.1: (7.2.2, 7.2.3, 7.5.1, and 7.5.2). 7.6.2: (7.2.2, 7.2.3, and 7.5.1). 7.6.3: (7.3.2). 7.6.4: (7.3.1). Q.E.D. The conclusions of Proposition 7.6 are easily combined to yield the follolving corollary. 7.7. Under the assumptions of Proposition 7.6, there exists a COROLLARY unique extension o f f to E < 2 X , provided 5 satisjies a n y one of the following: 7.7.1. E = J , ( X ) f o r s o n z e n ( b y 7.6.1 and 7.6.2). 7.7.2. J ( X ) C E C C ( X ) ( b y 7.6.1 and 7.6.3). 7.7.3. J ( X ) C G , and every E E 2 X has a jirst ( < r ) element ( b y 7.6.2 and 7.6.3). The final proposition of this section gives the number of different selections that may exist from a subcollection of 2 X to X . PROPOSITION 7.8. Let X be connected. Let k stand for the m a x i m a l number of different selections that can exist f r o m a jixed subcollection E of 2 X to X . T h e n : 7.8.1. I f J 2 ( X ) C E C C ( X ) ,then k=O or k 2 2 . 7.8.2. I f E > J ( X ) , or if E = J , ( X ) for some n, then k S 2 . Proof. 7.8.1: (7.2.4, and 7.6.1). 7.8.2: (7.2.4, and 7.6.3 (or 7.6.4)). Q.E.D. We conclude this section by giving some examples of spaces which do, (I2) .A different criterion for the existence of a selection on 2" or C ( X )is given in Proposition 8.3. 178 ERNEST MICHAEL [July or do not, satisfy condition 1.9.2 and/or 1.9.4. We begin by observing that a linearly ordered space, which is connected in the order topology, becomes disconnected by the removal of any point except, possibly, the two end points. Hence RPL(that is, Euclidean n-space), { x € R n llxll5 1 } , and {x€Rn( = 1 ) do not satisfy either 1.9.2 or 1.9.4 for n 2 2. The real line R1satisfies 1.9.4; it does not satisfy 1.9.2, as is easily seen by looking a t a compact neighborhood of that point of R1 which would have to become the first point in the ordering. Finally, a subset of R1 which has either a first or a last element in the natural ordering satisfies both 1.9.2 and 1.9.4. 8. More on selections. We shall again assume that ( X , T) and ( Y, T') are Hausdorff spaces, and that 2X and 2Ycarry the finite topology. DEFINITION8.1. We shall call X an (S1/S2/S3/S4)space, if there exists a selection from (2X/e(X)/an arbitrary disjoint covering of X by closed sets/an arbitrary disjoint covering of X by compact sets) t o X . 1 //XI/ LE1lnl.k 8.2. Let f : X -+ Y be continuous, open, closed, and onto. Then, if B C 2 is such that there exists a selection h from fV1**(S)to X , then there exists a selection from G to Y. Proof. The function f 0 h 0 f-I** is such a selection (see 5.10.2). Q.E.D. PROPOSITION 8.3. Let f : X -+ Y be continuous, open, closed, and onto. Then: 8.3.1. If X is an S , space, then Y is an S , space ( i = 1, 3). 8.3.2. If X is a n S , space, and if f-l(y) is compact for all y E Y, then Y is a n Si space ( i = 2, 4). Proof. This follows from Lemma 8.2 and, for 8.3.2, from Lemma 5.18. Q.E.D. The remaining results of this section deal with S3 and S 4 spaces. They are of a preliminary nature, and are stated without proof. LEMMA 8.4. If X contains a subset which is homeomorphic to a circle, then X is not a n S4 (and therefore not an S3) space. DEFINITION8.5. 8.5.1. A tree is a partially ordered set X , such that { t l t ~ x is) simply ordered for all x E X , and such that any two elements of X have a g.1.b. 8.5.2. A branch point of a tree X is a point x E X which is the g.1.b. of two distinct points of X , both of which are different from x. 8.5.3. The partial order topology P on a tree X is generated by sub-basic open sets of the following two kinds: (1) MY <sl, x E x, {yI y i s n o t z s ) , x E X. LEMMA8.6. Suppose that (X, T) can be partially ordered as a tree, such that the order topology P is coarser than T, and such that X has only afinzte number (2) 19511 179 TOPOLOGIES O N SPACES O F SVBSETS of branch points. Then: 8.6.1. X is an Sq space. 8.6.2. If,in addition, every closed ( T ) linearly ordered subset of X has a last element, then X is an Sa space. \Ve conclude by observing that a finite tree, in the sense of homology theory (for instance a closed figure X in the plane), satisfies 8.6.1 and 8.6.2. Unless this tree has only two vertices, however, it will not satisfy conditions 1.9.2 or 1.9.4 of Theorem 1.9. Appendix. We begin by re-examining the foundations of the finite topology. Consider a topological space (X, T). I t is easily seen (see [7] and Definition 1.8a, $2) t h a t the finite topology on &(X) (see Definition 5.1) is the join (or sup), in the lattice of all topologies on d ( X ) , of the following two topologies. DEFINITION 9.1. The upper (lower) semi-finite topology on d ( X ) is generated by taking as a basis (resp. sub-basis) for the open collections in d ( X ) all collections of the form ( E E ~ ( x ) E C U }(resp. f E E ~ ( x ) E n U Z @} ) with U an open subset of X . (To restrict these topologies t o 2=, me merely replace &(X) by 2X in the above.) I t is clear that these topologies (as well as their meet in the lattice of all topologies on d ( X ) , which we shall call the hemi-semi-finite topology) are in general not TI, and that they are admissible with respect to the generating topology on X. If X and Yare topological spaces, xire call a function f : X -+ d ( Y ) upper (lower) semi-continuous i f f is continuous with the upper (lower) semi-finite topology on d ( Y ) . This condition on f is equivalent to upper (lower) continuity of f in the sense of [9, pp. 148-1491, and to the strong upper (lower) semi-continuity, in the sense of [5, pp. 70-711, of the relation R(X, Y) defined by R(x, y ) a y Ef(x). The condition must be distinguished from another condition, which sometimes bears the same name, defined in terms of the lim sup (lim inf) of sets in Y or o f f (see [13, p. 1481, [9, pp. 104-1051, [5, p. 601, and [4, $I] which gives the simplest definition). This latter condition cannot, in general, be interpreted as continuity with respect to some topology on & ( Y ) . The connection between the conditions is established in [4, $I]. The following theorem follows from the definitions. / / THEOREM 9.2. If X and Y are topological spaces, then a function f : X Y) is upper (lower) semi-continuous if and only if f x E X ]f ( x ) n #~ 0 ] is closed (open) in X whenever A is closed (open) in Y. -+ &( I t is clear that f is continuous with the finite topology on &(I7) if and only iff is both upper and lower semi-continuous. We therefore have the following corollary. 180 ERNEST LZICHAEI, [July COROLLARY 9.3. I f X, Y a r e topological spaces, then a function f : X -+ d (Y ) continuous with the jinite topology on & ( Y ) i f and only if f x ~ ~ l f ( x ) n A # @ ] i s closed in X whenener A i s clo~edzn Y,and i s open i n X whenezler A 1s open in E'. 1s Theorem 9.2 and Corollary 9.3 give useful criteria for the coiiti~luityof a function into a hyperspace. Corollary 9.3 should have been stated a t the beginning of $5; several subsequent proofs could then have been simplified (5.10.2, for instance, follows directly from Corollary 9.3). T h e author, who mas not aware of this when $5 was written, apologizes to the reader for the inconvenience which this omission may have caused him. For the special case where f ( x l ) n f ( x z \ = @ for xl#xzl Theorem 9.2 yields the following improvement on 5.10.2. COROLLARY 9.4. W i t h the notation of 5.10.2, f-l* i s upper (lower) semicontinuous i f and only i f f i s closed (open). \ITe close this survey of semi-continuous functions with the following curious result which is essentially due to [5, p. 701. PROPOSITIO~ 9.4. I f X isjirst countable at xo, if Y i s completely regular, and i f f : X -+d ( Y ) i s upper semi-continuous at xo, then there exists a compact subset C of f ( x o ) with the following property: I f V i s a n open set containing C , then there exists a neighborhood U of xo such that f ( x ) C V U f ( x 0 )for a n y x E U. T o prove this proposition, look carefully a t the sets A,= C1 ( U f ( x ) ) n f ( x o ) , where the union is taken over all xESli,(xo) ( n = 1, 2, . . ). Let us now see how some of the results in $2 can be strengthened by means of our new topologies. Proposition 2.7, for instance, remains true if the finite topology is replaced by any of the above weaker ones. (Even the hemi-semi-finite topology ma\- be strictly finer than the factor topology on 2 ; see Example 2.10.) The same is true of Proposition 2.8, if we demand that every element of 23 should be connected. The most useful new result, however, is the follo\ving theorem. T I I E O R E9.5. ~ ~ Assertions 2.5.1 and 2.5.2 of Theorem 2.5 remain true (with unchanged proofs) if the finite topology i s replaced by the upper semi-finite topology. COROLLARY 9.6. I f X i s a compact space, Y a regular (topological) space, i f f :X -+ d(Y)i s upper semi-continuous, and i f f ( x ) i s closed (compact) irt Y f o r all x E X , then U Z E x f ( x )i s closed (compact) in Y. The last two results provide a systematic method for attacking various problems. \17e give two examples: 1. Lemma 5.18 follo\vs immediately from Corollary 9.6 and one part of Corollary 9.4 (see the remark follo\ving the lemma). 19511 TOPOLOGIES ON SPACES OF SUBSETS 181 2. I n 181, the proof of (3) in Theorem 1 is reduced to proving the follo\ving assertion: If X is a compact Hausdorf space, Y a topological space, F a collection of contznuous functions from X into Y which is compact in the compact-open topology, and if A is closed i n Y, then UfEpf-l(A) is closed i n X . This assertion follows a t once from Corollary 9.6, when we observe that the mapping q5 :F -+ 2X, defined by +Cf) =f-I(A), is upper semi-continuous. Finally, let us glance a t the uniform case. If [X, U ] is a uniform space with index set A, then 2U on d ( X ) is clearly the join, in the lattice of all uniform structures on d ( X ) , of the following two uniform structures. DEFINITION9.7. The upper (lower) semi-uniform structure on d ( X ) is generated by the index set A, and the neighborhoods U,(E) = ( F/ FC V,(E) ) (resp. TB,(E) = ( F ~ E C V , ( F ) ) ) for EE&(x). These uniform structures are clearly not TI, in general, and they are admissible with respect to U. \Ve call the corresponding topologies the upper (lower) semi-uniform topologies. I t is clear that the upper (lower) semi-uniform topology is coarser (finer) than the upper (lower) semi-finite topology. Furthermore, it follo\vs from Lemma 3.2 that the respective topologies agree on C ( X ) ; this generalizes Theorem 3.3. Much of what we have said for the finite case has its analogue in the uniform case. (Proposition 9.4 does not.) In particular, we have the following theorem. THEOREM 9.8. Assertions 2.5.1' and 2.5.2' of Theorem 2.5 remain true if the uniform topology is replaced by the upper semi-uniform topology. If X is a uniform space, then Theorems 9.8 and 9.6 make equivalent assertions about collections of compact sets; for collections of closed sets, however, the assertion in Theorem 9.8 is the stronger one. Finally, it is interesting to note that, in our applications, Theorem 2.5 itself was sufficient for the uniform topology, while for the finite topology we needed the improvement in Theorem 9.6. 1. N. Bourbaki, Topologie generale, Paris, Hermann 1940, chaps. 1 and 2. 2. , Topologie generale, Paris, Hermann, 1942, chaps. 3 and 4. , Topologie generale, Paris, Hermann 1948, chap. 9. 3. 4. R. Brissac, S u r les fonctions multiformes, C. R. Acad. Sci. Paris vol. 224 (1947) pp. 92-94. 5. G. Choquet, Conzlergences, Annales de I'UniversitC de Grenoble vol. 23 (1947-1948) pp. 55-112. 6. S. Eilenberg, Ordered topological spaces, Amer. J . Math. vol. 63 (1941) pp. 3 9 4 5 . 7. 0. Frink, Topology i n lattices, Trans. Amer: Math. Soc. vol. 51 (1942) pp. 569-582. 8. D. Gale, Compact sets of functions and function rings, Proceedings of the American Mathematical Society vol. 1 (1950) pp. 303-308. 9. H. Hahn, Reele Funktionen, Akademische Verlagsgesellschaft, Leipzig, 1932. 10. F. Hausdorff, Mengenlehre, 3d ed., Springer, Berlin, 1927. 182 ERNEST MICHAEL 11. J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. vol. 52 (1942) pp. 23-36. 12. C. Kuratowski, Topologie I, Monografje Matematiczne, Warsaw, 1933. , Les fonctions semi-continues duns l'espace des ensembles fermes, Fund. Math. 13. vol. 18 (1932) pp. 148-160. 14. L. Vietoris, Bereiche Zweiter Ordnung, Monatshefte fiir Mathematik und Physik vol. 33 (1923) pp. 49-62. 15. ---, Kontinua zweiter Ordnung, Monatshefte fiir Mathematik und Physik vol. 33 (1923) pp. 49-62. 16. A. Weil, Sur les espace a structure uniforme et sur la topologie generale, Paris, Hermann, 1937.