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M. Schroder, Analogues of the Compact-Open Topology, P 83

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M. Schroder, Analogues of the Compact-Open Topology, P 83 ANALOGUES OF THE COMPACT-OPEN TOPOLOGY M. Schroder (received 1 December, 1978; revised 10 December, 1979) Introduction For any collection L of subsets of a set X and any uniform space Y (indeed, for any uniform convergence space), the function space Y^ can be equipped with uniform L-convergence. It can also be given the L-open topology (when Y is topological), a topology shown to involve nothing more than the convergence of filters to sets. This idea enables one to define an L-convergence on Y^ for any convergence space Y at all, and then to recapture the L-open topology as a special case. Now suppose that X is a topological space and that C is a collection of continuous functions from X to Y . Classically, one finds (i) that if X is compactly generated and Y is topological, then the compact-open topology coincides on C with continuous convergence, (ii) that if X 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 topology coincide on C when Y is a uniform space. More generally, let C consist of all f in Y^ , such that the same filters converge to f under both L-convergence and uniform L-convergence. Then X becomes a compactly generated space, and (a) each member of L is compact, (b) C is the set of all continuous functions from X to Y , and (c) L-convergence, uniform L-convergence and continuous convergence all coincide on C , provided that L and Y satisfy certain mild restrictions. This extension and partial converse Math. Chronicle 10(1981) 83-98. 83 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. For any set Q , let Pow(Q) and Fil(Q) stand for the power set of Q and the set of all filters on Q . As usual, if A c Q and x € Q then A and x are the principal filters based on {i4} and {{as}} respectively. In particular, 0 = Pow(Q) is regarded as a filter, known as the improper filter. All other filters are proper. If M and N are collections of subsets of Q , one says that M meets W if M fl N is non-void. Every function h : Pow(P) — *■ Fil{Q) such that li(A U fl) = h(A) fl h(B) for all A , B in Pow(P) , can be 'extended' to a fl-homomorphism from Fil{P) to Fil(Q) as follows: if A is a filter on P , then h(A) is defined to be the filter U{h(i4) : A € A}. Lemma 1. Under these conditions, an ultrafilter on Q finer than h(K) is finer than h(U) for some ultra filter U finer than A . With each non-void collection r of filters on Q are associated its segmental, Choquet and principal modifications, denoted by oT , oT and nr respectively. A filter belongs (i) to oT as soon as it is finer than some member of r , (ii) to oF provided that every finer ultrafilter belongs to oT , and (iii) to nT if it is finer than the fi Iter nr = [A : A € A for all A f f). 84 Naturally r is said to be a segment if r = oT , Choquet if T = oT and principal if T = ttT . (There is another procedure known as solidification [8] which also leads to oT : so r may be called solid instead of Choquet if F = oT .) 2. Point and set convergence. One method of formalising the idea of convergence uses a function, associating with each point, the collection of filters which 'converge' to that point. Thus a (point) convergence y on Q is defined to be a function y from Q 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 x in Q , (^p) y(®) is a segment, and (Cj) x belongs to y(x) . Similarly, a set convergence on Q is a function from Pow(Q) to Pow(Fil(Q)) such that for all A , B in Pow(Q) and all filters F , G and H on Q , (SQ) r(4) is a segment, (Sj) A belongs to r(>4) , and (S2) if W is finer than F O G and F € r(>i) and G €r(B) then H £ r(4 U B). One often uses more descriptive terms such as "F — ►A under F", "x is a Y-limit of G " or the like, instead of "F € r(/l)" and "G £ y(*)" • Each set convergence T on Q defines a convergence F* by 'restriction' : F — *■x under r* iff F —> {x} under r . On the other hand, each convergence y can be 'extended' to a set convergence 85 y# as follows: F —A >- under y* iff ^ is finer than the filter fl{F : x £ A] , for some choice of the filters F in y(x) • x x However, not every set convergence arises in this way. For example, take an infinite set P , define y(x) = {$,x,U^} where the ^'s are distinct non-trivial ultrafilters, and put F in r(i4) if F — ►A under y* , but in the notation used above, F = x for all bar finitely x many x in A . In all cases though, y = y** and T is finer than r** (that is, T-convergence implies r**-convergence). Further, r = y* for some convergence y iff r = r** . Applying the operators o and it pointwise to the convergence y » one obtains its Choquet modification ay and its principal modification ny . Similarly one calls yChoquet or principal when the segments 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 belongs to every filter y-converging to some point of U . The set of all y-open subsets of Q is a topology, which generates a convergence xy coarser than y . Thus one calls y topological if y = xy . 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) The convergence y is principal iff yA is principal. (ii) If U is y-open and F converges to a subset of U under y* then U belongs to F . (iii) In particular, if y is topological then the y-neighbourhood filter of a set is the coarsest filter y A-converging to that set. 86 3. Uniform convergence structures. Again let e be a set, Q2 = Q * Q and Dq the diagonal in Q2 . C.ll. Cook and H.R. Fischer [3] called a collection ft of filters on Q2 a uniform convergence structure if (f/Q) ft is a segment, (U.) D belongs to ft , 1 y (U^) F O G belongs to ft if both F and G do, (#3) ft is symmetric, meaning that it is closed under converses, and (i/^) ft is closed under composition of relations. More generally, one calls ft a uniform rule if UQ to U 3 all hold: the yet weaker axioms used by some other authors would cause minor inconvenience later. The convergence w derived from a uniform rule ft by defining "F converges to x under a)” if F * x belongs to ft , is symmetric, in that y — *-x under w exactly when uj({/) = w(a:) . The properties U^ are all stable under Choquet modification, meaning that if ft satisfies U^ then so does oft . Furthermore, if ft is a uniform rule, the convergence derived from oft coincides with od) , and in particular, w is Choquet whenever ft is a Choquet uniform rule. 4. Compactness and regularity. Let y be a convergence on 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. Only a rather weak regularity axiom is needed here: y is called R j i if no proper filter both y-converges to x and rry-converges to y, 87 unless y — *■ x under y • (In [8], a slightly stronger axiom,R^ was needed to prove that any compact symmetric R^ ^ Choquet convergence was topological, and hence regular. Similarly, one can show that any compact symmetric R^ j Choquet convergence is principal: the existence of compact Hausdorff principal convergences which are not topological shows that R^ ^ is strictly weaker than R^ ^ .) 5. Multiplication and covers. Let y and 6 be convergences on Q . One constructs convergences y*6 and y*6 by demanding (W*) F — x under y*S iff filters G in 6(x) and R in y(j/) for all y in Q can be U so chosen that F is finer than fl{R : y $ G) , y for all G in G , and (M-:) F — +x under y*6 iff there is a filter G in 6 (as) such that F — ►G under y* for all G in G . Clearly y*6 is finer than y*6 and equally clearly, they coincide if y is principal.
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