Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
ANALOGUES OF THE COMPACT-OPEN TOPOLOGY M. Schroder (received 1 December, 1978; revised 10 December, 1979) Introduction L of subsets of a set For any collection space Y space Y^ X and any uniform (indeed, for any uniform convergence space), the function can be equipped with uniform L-convergence. Y given the L-open topology (when It can also be is topological), a topology shown to involve nothing more than the convergence of filters to sets. idea enables one to define an L-convergence on space Y Y^ This for any convergence at all, and then to recapture the L-open topology as a special case. X Now suppose that is a topological space and that collection of continuous functions from finds X (i) that if X Y . is compactly generated and then the compact-open topology coincides on convergence, to X (ii) that if C C is a Classically, one Y is topological, with continuous is locally compact and Y is a uniform space then continuous convergence and the topology of compact convergence coincide on C , and (iii) that compact convergence and the compact-open C topology coincide on Y when is a uniform space. More generally, let C consist of all same filters converge to f under both L-convergence and uniform L-convergence. functions from X Then (a) each member of X L to f in Y^ , such that the becomes a compactly generated space, and is compact, Y , and (b) C is the set of all continuous (c) L-convergence, uniform L-convergence and continuous convergence all coincide on satisfy certain mild restrictions. C , provided that L and This extension and partial converse Math. Chronicle 10(1981) 83-98. 83 Y of (iii) above is matched by similar improvements to (i) and (ii): the main problem left is - how much can the restriction be relaxed? Background Except for the basic calculus of filters, most tools used later are briefly introduced in this section: fuller treatment can be found in the works cited, results being stated mainly without proof. 1. Sets and filters. Q for the power set of if A c Q on {i4} x € Q and and {{as}} Q , let For any set Pow(Q) A x and respectively. that M If meets M and W if Every function N for all filter on Lemma 1. h(K) P , then h(A) in h(U) Q , one says Pow(P) , can be 'extended' to Fil(Q) as follows: if Q r of filters on Q nr respectively. A filter belongs finer than some member of ultrafilter belongs to r , oT , and (ii) to (i) to oF (iii) to oT = [A : A € A for all 84 oT , oT as soon as it is provided that every finer nT if it is finer than the fi Iter nr A . are associated its segmental, Choquet and principal modifications, denoted by and is a finer than for some ultrafilter U finer than With each non-void collection A U{h(i4) : A € A}. is defined to be the filter Under these conditions, an ultrafilter on is finer than All other filters such that A , B Fil{P) to a fl-homomorphism from is is non-void. h : Pow(P) — *■Fil{Q) li(A U fl) = h(A) fl h(B) As usual, 0 = Pow(Q) In particular, are collections of subsets of M fl N stand are the principal filters based regarded as a filter, known as the improper filter. are proper. Q . and the set of all filters on then Fil(Q) and A f f). Naturally r is said to be a segment if and principal if T = ttT . solidification [8] which also leads to solid instead of Choquet if 2. r = oT , Choquet if T = oT (There is another procedure known as oT : so r may be called F = oT .) Point and set convergence. One method of formalising the idea of convergence uses a function, associating with each point, the collection Thus a (point) convergence of filters which 'converge' to that point. y Q on y is defined to be a function Q from to Pou(Fil(Q) ) . Though other more (or less) restrictive definitions have been given elsewhere (see [5] or [6]), it is assumed here that for all (^p) y (Cj) x (®) is a segment, belongs to Similarly, a set convergence on Pow(Fil(Q)) G and H on Q A , B such that for all in Q , and y(x) . Pow(Q) is a function from Pow(Q) in and all filters to F , Q , (SQ) r(4) (Sj) A (S2) if is a segment, belongs to W r(>4) , and is finer than F € r(>i) F O G G € r(B) and then and H £ r(4 U B). One often uses more descriptive terms such as "x x is a Y-limit of G " or the like, instead of "F — ► A under "F € r(/l)" F", and "G £ y(*)" • Each set convergence 'restriction' : F — *■x T under other hand, each convergence Q on r* y defines a convergence iff F — > {x} under F* r . by On the can be 'extended' to a set convergence 85 y# F — >- A as follows: under y* iff ^ is finer than the filter fl{F : x £ A] , for some choice of the filters x F x in y(x) • However, not every set convergence arises in this way. take an infinite set P , define y(x) = {$,x,U^} distinct non-trivial ultrafilters, and put under F y* , but in the notation used above, many x in For example, where the in r(i4) F = x x ^'s are F —►A if for all bar finitely A . In all cases though, y = y** and T-convergence implies r**-convergence). convergence y iff T is finer than Further, r** r = y* (that is, for some r = r** . Applying the operators o and it pointwise to the convergence one obtains its Choquet modification y ay y » and its principal modification Choquet or principal when the segments ny . Similarly one calls y(x) are all Choquet or principal. Analogous terminology is also used for set convergences. Finally, a subset U of Q is called y -open if U . every filter y-converging to some point of subsets of than y . Q U belongs to The set of all y-open is a topology, which generates a convergence Thus one calls y topological if y = xy . xy coarser More information about topological convergence can be found in [5] and [7] along with references to some of the original work. Note though the following facts. Lemma 2. (i) (ii) The convergence If U is y-open and under y* (iii) y then is principal iff y A is principal. F converges to a subset of U belongs In particular, if y to F . is topological then the y-neighbourhood filter of a set is the coarsest filter yA-converging to that set. 86 U 3. Uniform convergence structures. Dq and Q2 . the diagonal in collection (f/Q) ft is a segment, (U.) D belongs to (U^) FOG 1 (#3) y ft e Q2 = Q * Q be a set, C.ll. Cook and H.R. Fischer [3] called a Q2 ft of filters on Again let a uniform convergence structure if ft , ft if both belongs to F G and do, is symmetric, meaning that it is closed under converses, and (i/^) ft is closed under composition of relations. ft a uniform rule if More generally, one calls hold: UQ to U3 all the yet weaker axioms used by some other authors would cause minor inconvenience later. The convergence "F converges to in that y — *- x x if with ft derived from a uniform rule under under The properties meaning that if w U^ w F * x a)” if exactly when belongs to ft by defining ft , is symmetric, uj({/) = w(a:) . are all stable under Choquet modification, satisfies U^ then so does oft . Furthermore, ft is a uniform rule, the convergence derived from od) , and in particular, w is Choquet whenever oft ft coincides is a Choquet uniform rule. Q . A subset K of Q is said to be compact (or more precisely, y-compact) if every ultra-filter on Q to which K belongs y-converges to some point of K , and y itself is called compact if Q is y-compact. 4. Compactness and regularity. Let y be a convergence on Only a rather weak regularity axiom is needed here: Rj i if no proper filter both y-converges to 87 x y is called and rry-converges to y, y — *■x unless under (In [8], a slightly stronger axiom, R^ y • R^ ^ Choquet convergence was needed to prove that any compact symmetric was topological, and hence regular. Similarly, one can show that any R^ j Choquet convergence is principal: compact symmetric the existence of compact Hausdorff principal convergences which are not topological R^ ^ shows that 5. is strictly weaker than Multiplication and covers. One constructs convergences (W*) F — x and R Let y*6 under in U y*S G (M-:) F —+ x in 6 (as) G all Clearly if and y y*6 y G in Q in is finer than y*6 Q . 6(x) can be fl{R y :y $ G) , iff there is a filter such that in be convergences on G , and F —► G under y* G for G . is finer than is principal. t under 6 by demanding for all F in and y*6 iff filters y(j/) so chosen that for all y and R^ ^ .) y*6 and equally clearly, they coincide Elementary properties of • were covered in [7], seems to behave similarly (no hidden significance attaches to the division sign). As was said to be diagonal (for reasons given in [5] y and [7]), it makes sense to call strongly diagonal if y This has a close connection with topology, since y y = yiy . is topological iff it is principal and diagonal [5 ]. For some purposes however, one wants y to be diagonal, not every where but only on a set or collection of sets. collection of subsets of induced by the inclusions UM then G —►x y(M)(x) = {0,x} under y and M Q and let M c Q . y(M) To be precise, if and otherwise, meets So suppose G — »- x G fl x , so that 88 M is a be the final convergence x under lies outside y(M) x € M € G iff for some M in M . that One then says that generates M y covers M UM = Q if y = and filter with a base in and the same y - limit, and that M y = yy(M) compactly generated if they generate In y , and y . Convergence in function spaces P Let N and P , Q of and Q R = be sets, and R . Given subsets L , M and respectively, one defines N(L) {/ (ac) : / ( N = [L,M] = x t L) , and { / € / ? : /(L) c M } as usual, and notes that N(L ) c M (*)... Further, for any filters filter on Q based on 9 on subsets of iff R N c [L,M ] . and (T(F) : J1 ( 0 Npw take a set convergence 0 —►/ r on F L : F iff 6(6) — ► f(L) on and Q P , let 0(F) be the f} . and a collection P , and define a convergence on under R under L of by demanding that r for all L in L . Basic properties of this type of convergence are listed below without proof. Theorem 3. (i) (ii) In the notation used above3 both P : r , is M-diagonal y y - y*y(M) . or is locally compact if the y-compact sets cover y y(M) if for each y-convergent filter there is a coarser or strongly M -diagonal according as particular, y {0} : r and 0 : r are the indiscrete convergences is pointvise convergencet if singletons, 89 P is the set of all K : (iii) L : r , if is coarser than r on the other hand, if (iv) K c L, is the closure of L under finite unions then K K : r = L : r , and if i = UL. then a filter converges under L : r i ^ iff it converges to the same limit under each (v) L. : r (that is, L : T is the supremum of the l- :.!*)• The consistency condition S2 is needed only in proving (iv) above: SQ the other claims hold for set convergences satisfying If Lemma 4. Proof. r is principal, so is Suppose —►/ under L : r . L : r . be the coarsest filter T-converging to than G Lj . Then by (*) , S j alone. and For each L in L let i\>(L) f ( L ) , so that G^ is finer i|/ itself is finer that the filter 6„ j generated by {[ L,M] : M t Gl a filter which clearly converges to and / L ( L) , under L : r , as desired. It is now only a short step to the compact-open topology. a convergence y respectively. Then on Q , and subsets [L,Af] [L,Af] , and that M is y-open. f(L) c M . By lemma 2, belongs to \p . Theorem 5. Let y is open in ip — *■f prove this, suppose that M L M of P and {£} : y* if M is y-open. and (L) : y * , that under i|<(L) — *■f(L ) Then belongs to be topological. ip(L) Then and so by L : y* f under Q To belongs to y* and (*) , [L,M] is also topological being the convergence derived from the L-open topology. 90 Consider Proof. i , so that in L : y* By theorem 3, Y-open. [£,M] is open in L : if L f for L M is L and is in As the neighbourhood filter of a function topology is generated by sets of the form M {L } : y* is the supremum of the in the L-open f(I>) c M [L,M] , where is y-open, it coincides by lemma 2 with the filter 0^, and defined in the proof of lemma 4. Consider now a uniform rule Alongside L : w* definition [4, §7], 0 — *• / filter based on (0 x (T x {/})(Z?L ) for T in Q fl on whose convergence is one also finds its uniform counterpart L : fl under f(x)) : g £ T = 0 , belongs to fl . L iff for each at . L : fl . in By L , the x f L] and Even in the simplest case of a metric space such as the real line, the L-open topology and the topology of L-convergence do not always coincide, and may indeed be incomparable. Naturally one asks (i) when does L : fl-convergence, and L : id* convergence imply (ii) when does the converse hold? Partial answers are given in the next section. First comparisons Thoughout this section, let Q P , fl be a uniform rule on from P functions by Q , and to h and h(A) = 0(i4) k and 0 L be a collection of subsets of whose convergence is be a proper filter on from Pow(P) k(A) = to (0 x ^ ( Z ^ ) Fil(Q) u> , / p E = Q . Then the Fil(Q2 ) , defined and for all A in Pow{P) , satisfy the condition laid down in lemma 1 . Suppose first that (Ag) each 0 —►/ f(L) under L : , and that lies inside an w-compact set, 91 be a function (Tj) L has the finite intersection property, (X 2 ) if F G Proof. under ft is Choquet, and (V2) u> is strongly compact-diagonal. 0 —►/ L Take in under L : ft . So, let U By lemma 1, there is an ultrafilter By XQ , its image K f(L) . If /(ti) 0(U) 3 0 (A/) R ft , or be an ultrafilter 9 (M) — ► f(M) This means that 0(U) — *- y y under under under w y in a w-compact set X^ . by u>* , for all w * aj({A}) R ^ k(U) . such that U fl L , then 1/ is the filter based on and L converges to some point /(V) , a filter converging to K belongs to Further, M U fl L . in and hence under w , by . But to belongs to k(L) = k(L) . finer than 72 k(L) L , and try to show that V ^ , that every finer ultrafilter does. by w , and w , (1^) Then under F fl L , then is the filter based on /(G) — *■y Lemma 6. L , /(F) — > y meets ft k(U) is clearly finer than since both 0 (U) the still finer filter and R /(U) also belongs to On the other hand, suppose that (X3) each 0(U) x /(U) , a filter belonging converge to f(Q 0 —> / y under w . ft , as desired. under L : ft , and that is m-compact (X3) ft is a uniform convergence structure, and ( V o) is F ,.i and Choquet. 92 In short, Then Lemma 7. Proof. Q — *■f under L Again take in L : to* . L , let R h(L) , and "choose" an ultrafilter y , some point f(L) of be an ultrafilter finer than il L R ~> h(U) . such that f(U ) . is the to-limit of k(L) f(U) * f(U) belongs to h(U) x /(U) = k(U ) o {/(U) x f(U)} , as can be seen by comparing basis fl , as do and k(U ) . By Consequently Further elements: T(U) h(U ) Thus and X /([/) R y finer than f(L ) Though * G^ = 0 f(L) in h{L) Zy if G y Being finer than y —> x (because under w to under let to . In short, every f(L) . be the set of all ultrafilters y , and is void, in all cases y So, suppose y y converges to some point of G y G^ — be their intersection. y under : y t /(£)) , it converges to as desired, provided that each G . {/})(zy o (f(U) x f(U)) . x which w-converge to h(L ) = fl{G Further, as (T both converge to ultrafilter finer than For each = G^ — * y under R h(L) , it to-converges to some point (because to j) and so is — *■y Hence under u* , to . is proper, and take an ultrafilter is symmetric). nw . f(L) under R — *- y to finer than x . Thus under to (because to is C h oquet). Altogether, conditions (K) fl to are equivalent to is a Choquet uniform convergence structure whose convergence to is ^2 1 anc* stron8ly compact-diagonal. One should ask whether V forces generated by a classical uniformity. 93 fl to be principal, and hence Consider the following example. Q be the set of all complex numbers in the open right half Let plane, together with the origin. Qn and let and t T be the usual topology and uniformity on n . for some U fl , put and t Q and in fl Q . U ? T iff and Clearly a filter w-converges to a non-zero limit H — *• 0 iff it x-converges to that limit, while under n > 0 , set {x + iy € Q : \y\ < nx) , = To define the uniform rule Q^2 £ U For each integer G H , for some n . fl are the inductive limits of By construction, w under u> iff H — *■0 (In more technical language, T l#n and <d respectively.) is neither topological nor even principal - but it is c-embedded, [l, Theorem 33] and hence regular, and it is strongly diagonal (as one can see after a little calculation). fl Moreover, is a Choquet uniform convergence structure, but clearly not principal. ft and In all, ui satisfy V without being "classical". These results may make better sense when one sees that X 3 help define a convergence on gence y P . Q , and a collection °n X^ , X2 and To be precise, consider a conver L of subsets of P satisfying , and let C Clearly C By /(F) — ► f(x) X^ and and *3 hold} . contains all the constant maps. The convergence iff {/ € QP : X2 = B under X 3 , each every member of on C y L P is defined as follows: f for all in is 6-compact and is B-y-continuous. C , and 94 L meets L generates 8 . under H —►y B F fl x . Also, (A function is said to be continuous if it "preserves convergence", that is, 0 (H) — * g(y) •) F —> x implies The converse is true too, at least if g this, suppose L . meets G F Then 1/ converges to some point g(x ) = y But since x under X2 and under y , and F F fl L . U x L in W y V fl L based on y under is Choquet. in g(V) — ► g(x ) and converges to y , because L Choose a 8-compact set and the filter B , the filter g(G) — *- y To prove U o g(G ) , one can "find" an ultrafilter g{\J fl L) c (j . such that under be the filter based on Given an ultrafilter V is Choquet. g(T) — * y is B-y-continuous, Also, let y under y . converges to y . Hence g In all, (/ fl L . satisfies *3 . A global comparison of Theorem 8. Let satisfy L functions satisfying L : , ft satisfy X2 and coincide on L : w* L : fl and X3 . id* is now easily made. V s and C be the set of Then the convergences L : ft and C . Second Comparisons Next let a and Con(a,y) and let y be convergences on Despite its name, continuous convergence Con(a,y): , not just on under y whenever continuously to C j outside Theorem 9. diagonal. f F — >■x iff Con(a,y) . f Q and respectively, L : y* 0 — *■f under con(a,y) under a . can be defined on con(a,y) However as f con(a,y) is continuous, 0(F) — *■f(x) iff converges does not satisfy For more details, see [3], [5] or [l]. Suppose that Then P stand for the set of all a-y-continuous functions. L generates and that a y is strongly is finer than continuous convergence, on Con(a,y) . Proof: If G Let 0 — *■f under L : y* , and is the filter based on F F fl L , then 95 converge to /(G) -*■ /(x) x under under a . a , while 0(F) ^ 0(G) and Hence 0(F) — *• f(x) 0(C) —> f(G) under under y* , for all G in y is merely F fl L . y -’ y = y , as desired. From the proof, one can see that this remains true if strongly compact-diagonal, but the members of L are cx-compact. In the uniform case, the comparison below is well known. Theorem 10. cover a . Let ft be a uniform convergence structure on Then Q , and L L : ft is finer than continuous convergence, on Con{a,uj) . K . Next, consider {^}-convergence, for any a-compact set By methods similar to those of the previous section, one can prove that these are coarser than continuous convergence. Lemma 11. Let K be a-compacts and y be Then continuous convergence is finer than Lemma 12. Let K be a-compact, and continuous convergence is finer than Theorem 13. diagonal, Let y be R 1 1 , symmetric and Choquet. {K}:y*, on P Q . SI be a Choquet uniform rule. (if) : ft , on Then Con (a, to) . Rj l J symmetric, Choquet and strongly compact- SI be a Choquet uniform convergence structures and K the set of all a-compact sets. (i) If a is compactly generated then continuous convergence, on (ii) If a Theorem 14. If a More generally, if , then a(K) K : ft coincides with Con(a,w) . is compactly generated and continuous convergence, K : ft and K coincides with Con(a,y) . is locally compact then continuous convergences on K : y* ft satisfies K : ai* all coincide, on V s then Con (a, to) . is a family of a-compact sets, satisfying is a compactly generated convergence finer than 96 a . Thus if U V , then both satisfies con(a(K) ,u>) , on Con(a(K) ,w) K : ft and K : and in particular, on coincide with Con(a,y) . In fact, one can prove theorem 8 using these comparisons with continuous convergence, theorem 14, and the convergence 8 defined earlier; the proof is no easier, though. Several problems remain. can R^ j be weakened, say, to An obvious technical one concerns R^ V : (I guess not) or strong compact- diagonal ity to compact-diagonality (I don't see how)? Since the "constant function" map embeds y continuous convergence can be topological only if in y con(a,y) , is. In cases discussed by R. Arens and H. Poppe, say, [6, Theorems 2.8, 2.8(a) and 2.9], if con(a,y) is topological then compactly generated. L-convergence, then is: a is locally compact or con(a,y) Similarly, if what conditions on a coincides with is "often" compactly generated. a and y The problem lie behind this "often"? REFERENCES 1. E. Binz, Continuous Convergence on C(X), Lecture Notes in Mathematics 469, Springer, 1975. 2. E. Binz, H.H. Keller, Funktionenraume in der Kategorie der Limesraume, Ann. Acad. Sci. Fenn. Ser A1 383 (1966). 3. C.H. Cook, H.R. Fischer, On Equicontinuity and Continuous Convergence, Math. Ann. 159 (1965), 94-104. 4. C.H. Cook, H.R. Fischer, Uniform Convergence Structures, Math. Ann. 173 (1967), 290-306. 5. H.J. Kowalsky, Limesraume und Komplettierung, Math. Nachr. 11 (1954), 143-186. 6. H. Poppe, Stetige Konvergenz und der Satz von Ascoli und Arzela, Math. Nachr. 30 (1965), 87-122. 7. M. Schroder, Adherence operators and a way of multiplying convergence structures, Math. Chronicle 4 (1976), 148-162. 8. M. Schroder, Compactness Theorems, in Lecture Notes in Mathematics 540, Springer, 1976, 566-577. University of Waikato 98