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Carnegie Mellon University Research Showcase @ CMU Department of Mathematical Sciences Mellon College of Science 1967 Natural covers S. P. Franklin Carnegie Mellon University Follow this and additional works at: http://repository.cmu.edu/math This Technical Report is brought to you for free and open access by the Mellon College of Science at Research Showcase @ CMU. It has been accepted for inclusion in Department of Mathematical Sciences by an authorized administrator of Research Showcase @ CMU. For more information, please contact [email protected]. NOTICE WARNING CONCERNING COPYRIGHT RESTRICTIONS: The copyright law of the United States (title 17, U.S. Code) governs the making of photocopies or other reproductions of copyrighted material. Any copying of this document without permission of its author may be prohibited by law. NATURAL COVERS by S. P. Franklin 67-29 September, 1967 NATURAL COVERS by S. P. Franklin §0. INTRODUCTION. In recent years the category of has come to occupy a fairly central place in topology k-spaces (see for example Whitehead [20], Gale [15], Michael [18], Cohen [9], Morita [19], Arhangel1skii [l], Bagley and Yang [5], Duda [10], Whyburn [21], Arhangel1skii and Franklin [3] and many others). R. Brown suggests this category may serve all the major purposes of topology [8]. Of more recent vintage is the interest in sequential spaces (see for example Kisyn'ski [17], Dudley [11], Aull [4], Franklin [I3],[l4], Boehme [7], Baron [6], Arhangel1skii and Franklin [3], Fleischer and Franklin [12]). Several people have noted (and exploited) the similarity of the theorems which can be proved about spaces. k~spaces and sequential In this paper we offer a general theory which en- compasses the common aspects of the theories of and sequential spaces and of others also. k-spaces The ideas involved can be traced from Michael1s generating collections [18], thru Cohen's weak topologies [9] and Brown's natural covers [8]. We strengthen Brown's concept slightly, retaining his terminology. Section 1 is devoted to the statements of some preliminary facts and lemmas due either to Brown [8] or to Cohen [9] . The theory is then developed in sections 2 and 3. HUNT LIBRARY CARNEGIE-MELLON UNIVERSITY §1. THE WEAK TOPOLOGY OF A COVER. Let topology E be a cover for a topological space T . The family which intersect each S e £ in an with the relative topology from X finer than X S-open set (i.e. open in T ) The restriction of the same topology on with E ( T ) of those subsets of S T . X is a topology for £ ( T ) to S yields S (i.e., E(T)/S = T / S ) , and repetition of the process leads to no new open sets (i.e., £(£(T)) = S(r)) One key fact is given in the following. 1.1 LEMMA. A function out of X only if^ its restriction to each If we let the spaces 0 £ E(r)~continuous if and S e £ ijs continuous. denote the disjoint topological sum of S e £ , the inclusion maps yield a quotient map for a space Xf <p: ^ E—>.X . with topology T* is the associated quotient map. is a cover of 1.2 is X x Xf LEMMA^ with <p X <p» : i'+} E X Q S c x Suppose and that Then combine to S! is a cover <p! : ©EI~>XI . X X £' = (S X S'|S e S, S and X X Xf topologized by S1 — > X X X ' (S X ZJ1 ) (T X T ! ) , is a quotient map. Further we have 1.3 LEMMA. E ( T ) x S' (T» ) c (E x Ef ) (r x r) . is^ a^ £(r)~interior point of some each if: each x e X S e E ^ and similarly for xf e Xf , then equality holds. The next lemma will prove central to our theory as it allows the construction of a coreflexive functor. 1.4 KEY LEMMA. rf for each Sf e £f i) there is an is continuous,, then and S e £ , f : X -> Y f(S) 5 ST with f and L 5 X1 ±s^ continuous with respect to £ ( T ) or if £ £ and £' are covers of is a refinement of If A c X , let LEMMA. ^ ^^^ £(T) |A <== If Ef (r ) c £ ( T ) . £ ( T ) = £T (r) . £|A = {A 0 s|S e £} , and let be the relative topology on 1.5 X . £? , then Hence if each is a refinement of the other G ii) f |S:S~>S' £'(T>). Now suppose both S satisfies T|A A . (£|A)(T|A) . Further, if. A T-clbs'edj then equality holds and A and each i^ E ( T ) closed. Refering to Example 5.1 of [13] , let E be the collection of all convergent sequences (with limit points) in A = M\N . Each € (£|A)(r|A) S e £ is closed but but not to and let is not, and {0} £ ( T ) |A . For another example^ let [0,1] A M I be the closed unit interval with the usual topology, and let of all connected subsets of be the collection Let A be the Cantor set . A , being compact, is closed but each S need not be. I G £ , T = £(r) . But is discrete. §2. r|A I . S is compact while (£|A)(T|A) Thus the conditions of lemma 1.5 are needed. NATURAL COVERS AND THEIR SPACES. By a natural cover we shall mean a function assigns to each topological space 1) Since if S e 2L and S X a cover £ IL is homeomorphic to a subset which satisfying T to Y , -then T e L S € £ , and 2) there is a may choose £v if f: X ~ * Y T G L with is continuous and f (S) 5 T . For example we to be the compact subsets of X , or the connected subsets, or the countable subsets, or the convergent sequences. The first and last of these are the notivating examples which lead to the Now to each space the space k~spaces and the sequential spaces X with topology T we may associate cxx , the same set of points topologized by We may also assign to each continuous function the continuous (by Lemma 1.4) function Since 0 of = f , a f : X—~>Y , f = erf : crx—^aY. . preserves compositions and identies, i.e., 1*L 1L f^nctor from the category \Tj of topological spaces into itself. Let us call a space (i.e. T = E (T)) . If X L a S~space whenever is the natural cover which assigns to each space its compact subsets, the the k-spaces. If E aX = X S-spaces are precisely assigns the convergent sequences, the £-spaces are the sequential spaces. (This is almost immediate.) 2.1 jls a, 2>space LEMMA. Hence a For each space X , aX is^ a, retraction from ^ (acrX = aX) . onto the category S-spaces. Proof. Clearly £X(T) ESaX(£X(r)) is a subspace of o*X ^x E Eax • E Hence * Now and hence belongs to a x ( E x ( r ) ) E hShS7^ if S G T *X > S = E v , i.e., x(r) and t h e lemma is proved. 2.2 LEMMA. S (g) i£ a. coreflexive subcategory of T) . Proof. lv : X — > X and is a Y is continuous. A f \ Y—>aX 1Y . 2.3 f : Y ~~^>X is continuous S-space the following diagram commutes. ^ ->X Y- Hence If r Y r T^^T^ * is continuous and uniquely factors f thru This completes the proof. PROPOSITION. The category (§} o£ S~spaces i£ closed under quotients and disjoint topological sums. Proof. This is a direct consequence of Lemma 2.3 and Theorem A of Kennison [16] since the disjoint topological sum is the coproduct in @ . It is not at all difficult to give a direct proof. 2.4 a 2.5 COROLLARY. Every open or closed image of _a &-space is TJ- space* COROLLARY. When a, product space i^s a^ £-space,, so is each of its factors. 2.6 COROLLARY. The continuous image of a compact in a Hausdorff space is ^ 2.7 COROLLARY. S-space 2>space. The inductive limit o«f any system o£ I^spaces is again ji 2>space. 2.8 COROLLARY. Any adjunction space of S~spaces is a Of course Proposition 2.3 and its corollaries may be immediately asserted for k-spaces and sequential spaces. E-space. 2.9 LEMMA. For each X , © E is, a, E-space. X Proof. If fore S e L S = OS . , then S 5 S Hence by 2.3 implies ±> £ x is a This allows a characterization of 2.10 PROPOSITION. natural mapping Proof. X 1+) IL quotients, if cp and there- E-space. E-spaces as follows. isi a. E-space if and only if the ^ x ^x <P : ® Since S e Sg is a — fL quotient mapping. S-space and is a quotient map, X S is closed under is a E-space. The converse has already been noted. Hence we get the sequential spaces as quotients of zero-dimensional, locally compact metric spaces and the kspaces as quotients of locally compact spaces. (Lemma's 1.2 and 1.3 together with Proposition 2.10 provide a criterion for the product of two be a E-space. E-spaces to again Although possibly useful in particular cases, it is clumsy to state and we shall omit it.) This characterization is in fact more useful than it would appear. For example, with it we can say something about subspaces of 22-spaces.. 2.11 If ^ € PROPOSITION. ^Y X and every open subset of each ifL IL E~space, then every open subset of X is a E-s-pace. Proof. Suppose U the natural mapping. open in <P~ (U) is open in X and Then for each S and hence is a is a E-space and E-space. <p : gi E y ^ X S e L , <p is (U) H s is Thus their topological sum ^ = <p|<p~ (U) maps (p^CU) onto U . By 2.10 it is enough to show For V 5 U , if ^ rl (V) ^ to be a quotient map. is open in ^""1(U) , then for each S e £ v , ^ rl (V) 0 S - ^"1(V) (1 (p^OJ) n S is open in A Since X is a ID-space, V is open in X and hence S . U , and we are done. From 2.11 one sees at once that open subsets of sequential spaces and k-spaces are again sequential spaces and It is also true that being a sequential space or a is a local property. 2.12 PROPOSITION. Let Ij[ each X and A e £ Y (r) x e X has an open neighborhood jLs a. S~space. choose x A open e A . x. . If S e L O is open in union of the U f|s f is continuous. The 2.13 L X . But , then A 3 r \ S - (T |U) | S A is the f : X—>Y %~continuous just in S G L . It follows from ±s^ S-continuous if and only if f : crx—>Y £-spaces can be characterized in terms £-continuous functions. PROPOSITION. X is; a. £~space ije and only if every ^-continuous function out of Proof. c S <= U e T and hence in is continuous for each Lemma 1.1 that of the be an A 0 U , and we are done. Let us call a function case U U ~-~ S fl (n 0 A) = Sfi A € T | S . But since A fi U Let O £-space neighborhood of Thus k-space This follows from which is _a S-space, then Proof. k-spaces. If each l x : X —~>aX Conversely X ^continuous function is continuous, then is continuous and f is continuous. Is continuous. being But X is homeomorphic to £-continuous implies that X = aX since X is a f : a-space. aX . 8 Following 3 we may assign an ordinal number to each in a topologically invariant manner. A a subset of X . Define A° = A;Att = (A^> let wise. The A Let X be any space and = U(ci(A n S)|S e S Y } . a = j8 + 1; A a = U{AP|jS < a] if S-space S-characteristic of Now other- X is the least ordinal a (if a it exists) such that for each subset A of X , A = c ^ x ^ ' (See [3] for the existance of sequential and Again we may characterize the 2.14 PROPOSITION. X i_s a k characteristics.) S-spaces. Z-space JLf and only if_ jU has a_ ^" characteristic. Proof. If X is a £~space, it suffices to show that for each OL A ^ X there is an ordinal a such that simply takes the sup of such ordinals.) A = cl^A . (One Clearly for each OL , A c: A ^ clYpA . If A is not closed, there is some S e ZL with A H s not closed in S , i.e. there is some point in AP \ AP . Hence by cardinality, some closed and hence equals clYA . Conversely if A X is has £- A characteristic OL and A fl S A then A is a £~space. ^A and hence A For any natural cover 0 is" closed in S for each c A and E , a if and only if it is discrete. spaces and of special study. A is closed. S-space has Thus ^characteristic < 1 (called kr-spaces respectively) have received We formalize the common part of these theories in the next section. ST-SPACES AND HEREDITARY QUOTIENT MAPS. A mapping X In the case of sequential k-spaces, those of characteristic Frechet spaces and §3. S e OL f : X—VY is an hereditary quotient map if and only if for each subspace YJ of Y , the mapping 9 f Arhangel1 skii showed that = f |f~~ (Y.j)is a quotient map. ° P e n maps and closed maps are hereditary quotient and characterized them as follows: a mapping f : X an hereditary quotient map if and only if for each for each open neighborhood point of X 3.1 f(U) U of_ f~ (y) , y y e Y and i^ ail interior ([2]). £T-space if its is called a PROPOSITION. If X Suppose A c y and f""1(y) 0 cl f"1(A) ^ 0 . L-characteristic is Sf - space and isa an hereditary quotient map, then Proof. >Y ±s^ f : X—^Y < 1 . i£ ijs JEI Sf-space. Y y e cl A . We claim that (If it were empty, U = X \ cl f"1(A) v -1 would be an open neighborhood of f (y) and by Arhangel 1 skii 1 s characterization of hereditary quotient maps, interior point of hence f (U) c Y \ A f(U) . E since c y e cl A.) Then for some Then f : S—>S T and Choose S eZ^ is a natural cover, there is some f(S) c s1 . would be an X \ cl f"1(A) <~X \f~1(A) contradicting x € f^Cy) n cl f"1(A) . Since But y , x € cl g (S fl f""1(A)) S'e L with y e f [cl Q (f 1 (A) n S) ] c cl Qf [f (f"1(A) 0 S) ] , is continuous. Thus y e cl e . (A flf(S)) clg, (A (1 ST ) and we are done. The preceeding proposition and the suceeding lemmas are aimed at characterizing E'-spaces in terms of hereditary quotient maps. 3.2 LEMMA. Every disjoint topological sum <o£ Sf - spaces is again _a Sf -space. 10 3.3 LEMMA. Ijf X € 2^ , then 3.4 LEMMA. Each 3.5 LEMMA. For each 3.6 PROPOSITION. S e ZL X i£ a S'-space. .is «* £'- space. L 1 -space. X , (& IL- is a i£ a. £f - space if_ and only jLf <P : X S is an hereditary quotient map. Proof. Proposition 3.1 and Lemma 3.5 yield one direction immediately. For the converse we will again use Arhangel*skii1s characterization. U Suppose X is an open neighborhood of <p~ (x) x € cl(A<p(U))) „ then for some Hence^ regarding contradicting S in U <p(U) x e X <£> 2L- . 5 and If S e Z^ , x e cl g (S n(xVp(U))) . as a subspace of cp~ (x) ^ interior point of Lf-space , is a and U IB 2L open. x e cl(SMJ) 5 Hence x is an and we are done. Since the Frechet spaces are precisely the hereditary sequential spaces ([14] Proposition 7.2), one might look for a similar result for any natural cover. Unfortunately., using the connected sets to generate the natural cover,, an example can be constructed of a not a £-space. S!-space with a subspace which is However something can be said in the general case. 3.7 ^ PROPOSITION. If each subspace of X isa £-space, then iiL -BL S* -space. Proof. Let tp : y £ T^~>X be the natural mapping, X^ be 1 a s u b s p a c e of X and <p = <p <p~ (X n ) . Let <pn : •+» J. U <= x and <p " 1 (U) is open in — o 1 1 <P ~ (U) = <P~ (U) n ±s ZL. is open in be the natural mapping. V (X ) , then 1 J. If O A, 11 But <p <p- is a quotient map and so is a quotient map and <p U is open in X^ . Thus an hereditary quotient map , and 3.6 completes the proof. 3.8 COROLLARY. rf every subspace o£ a. Z)r - space .is _a S-space,, then every subspace is a ST - space. REFERENCES [l] A. B. Arhangel1skii, Bicompact sets and the topology of spaces, Dokl. Akad. Nauk, SSSR, 150(1963), 9. [2] A. B. Arhangel1skii, Some types of factor mappings and the relation between classes of topological spaces, Soviet Math. Dokl., £(1963), 1726-1729. [3] A. B. Arhangel1skii and S. P. Franklin, Ordinal invariants for topological spaces, to appear. [4] C. E. Aull, Sequences in topological spaces, Not. Amer. Math. S o c , _12(l965), 65T-79. [5] R. W. Bagley and J. S. Yang, On k-spaces and function spaces, Proc. Amer. Math. S o c , 1/7(1966), 703-705. [6] S. Baron, The coreflective subcategory of sequential spaces, to appear. [7] T. K. Boehme, Linear s~spaces, Proceedings of the Symposium on Convergence Structures, University of Oklahoma, 1965. [8] R. Brown, Ten topologies for X X Y , Quart. J. Math., Oxford, (2) 14/1963), 303-319. [9] D. E. Cohen, Spaces with weak topology, Quart. J. Math., Oxford, (2) 5/1954), 77-80. [10] E. Duda, Reflexive compact mappings, Proc. Amer. Math. S o c , 17/1966), 688-693. [ll] R. M. Dudley, On sequential convergence, Trans. Amer. Math. S o c , 112(1964), 483-507. [12] I. Fleischer and S. P. Franklin, On compactness and projections, to appear. [l3] S. P. Franklin, Spaces in which sequences suffice. Fund. Math., 57/1965), 107-116. [14] S. P. Franklin, Spaces in which sequences suffice II, Fund. Math., to appear. [15] D. Gale, Compact sets of functions and functions rings, P r o c Amer. Math. S o c , 1/1950), 303-308. [16] J. F. Kennison, Reflective functors in general topology and elsewhere, Trans. Amer. Math. S o c , 118(1965), 303-315. [17] J. Kisynski, Convergence du type L , Coll. Math., 7(1959), 205-211. [•18] E. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. S o c , 1JL(1952) . [1.9] K-Morita, On decomposition spaces of locally compact spaces, Proc. Japan. Acad., 32(1956), 544-548. [20] J. H. C. Whitehead, Combinatorial homotopy, Bull. Amer. Math. S o c , 55/1949), 213-245. [21] G. T. Whyburn, Directed families of Sets and closedness of functions, Proc. Nat. Acad. Sci., U.S.A., 54(1965),. 688-692. CARNEGIE-MELLON UNIVERSITY PITTSBURGH, PENNSYLVANIA HUNT LIBRARY CONEfilMfELLON UNIVERSIh