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. Elementary properties of • were covered in [7], and t seems to behave similarly (no hidden significance attaches to the division sign). As y was said to be diagonal (for reasons given in [5] and [7]), it makes sense to call y strongly diagonal if y = yiy . This has a close connection with topology, since y 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. So suppose M is a collection of subsets of Q and let y(M) be the final convergence induced by the inclusions M c Q . To be precise, if x lies outside UM then y(M)(x) = {0,x} and otherwise, G —x »- under y(M) iff G — ►x under y and M meets G fl x , so that x € M € G for some

88 M in M . One then says that M covers y if UM = Q and y = y ( M ) , that M generates y if for each y-convergent filter there is a coarser filter with a base in M and the same y-limit, and that y is M-diagonal or strongly M-diagonal according as y = yy(M) ory - y*y(M) . In particular, y is locally compact if the y-compact sets cover y , and compactly generated if they generate y .

Convergence in function spaces

Let P and Q be sets, and R = . Given subsets L , M and N of P , Q and R respectively, one defines

N(L) = {/(ac) : / ( N and x t L) ,

[L,M] = {/€/?: /(L) c M }

as usual, and notes that

(*)... N(L) c M iff N c [L,M] .

Further, for any filters 9 on R and F on P , let 0(F) be the filter on Q based on (T(F) : J1 ( 0 and f} .

Npw take a set convergence r on Q and a collection L of subsets of P , and define a convergence on R by demanding that 0 — ► / under L : F iff 6(6) —f(L) ► under r for all L in L . Basic properties of this type of convergence are listed below without proof.

Theorem 3. In the notation used above3

(i) both {0} : r and 0 : r are the indiscrete convergences

(ii) P : r is pointvise convergencet if P is the set of all singletons,

89 (iii) K : r is coarser than L : r , if K c L ,

(iv) on the other hand, if L is the closure of K under finite unions then K : r = L : r , and

(v) if i = UL. then a filter converges under L : r i ^ iff it converges to the same limit under each L. : r (that is, L : T is the supremum of the l- :.!*)•

The consistency condition S2 is needed only in proving (iv) above: the other claims hold for set convergences satisfying SQ and S j alone.

Lemma 4. If r is principal, so is L : r .

Proof. Suppose — ► / under L : r . For each L in L let G^ be the coarsest filter T-converging to f(L) , so that i\>(L) is finer than G . Then by (*) , i|/ itself is finer that the filter 6„ Lj j generated by

{[L,M] : M t Gl and L ( L) , a filter which clearly converges to / under L : r , as desired.

It is now only a short step to the compact-open topology. Consider a convergence y on Q , and subsets L and M of P and Q respectively. Then [L,Af] is open in {£} : y* if M is y-open. To prove this, suppose that ip — *■ f under (L) : y* , that f belongs to [L,Af] , and that M is y-open. Then i|<(L) —f(L *■ ) under y* and f(L) c M . By lemma 2, M belongs to ip(L) and so by (*) , [L,M] belongs to \p .

Theorem 5. Let y be topological. Then L : y* is also topological being the convergence derived from the L-open topology.

90 Proof. By theorem 3, L : y* is the supremum of the L { } : y* for L in i , so that [£,M] is open in L : if L is in L and M is Y-open. As the neighbourhood filter of a function f in the L-open topology is generated by sets of the form [L,M] , where f(I>) c M and M is y-open, it coincides by lemma2 with the filter 0^, defined in the proof of lemma 4.

Consider now a uniform rule fl on Q whose convergence is at . Alongside L : w* one also finds its uniform counterpart L : fl . By definition [4, §7], 0 — *• / under L : fl iff for each L in L , the filter (0 x based on

(T x {/})(Z?L ) = f(x)) : g £ T and x f L] for T in 0 , belongs to fl . 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 : id* convergence imply L : fl-convergence, and (ii) when does the converse hold? Partial answers are given in the next section.

First comparisons

Thoughout this section, let L be a collection of subsets of P , fl be a uniform rule on Q whose convergence is u> , / be a function p from P to Q , and 0 be a proper filter on E = Q . Then the functions h and k from Pow(P) to Fil(Q) and Fil(Q2) , defined by h(A) = 0(i4) and k(A) = (0 x ^ ( Z ^ ) for all A in Pow{P) , satisfy the condition laid down in lemma 1 .

Suppose first that 0 — ► / under L : , and that

(Ag) each f(L) lies inside an w-compact set,

91 (Tj) L has the finite intersection property,

(X 2) if F meets L , /(F) — >y under w , and G is the filter based on F fl L , then /(G) — *■ y under w ,

(1^) ft is Choquet, and

(V2) u> is strongly compact-diagonal.

Lemma 6. Then 0 — ► / under L : ft .

Proof. Take L in L , and try to show that k(L) belongs to ft , or by V ^ , that every finer ultrafilter does. So, let R be an ultrafilter finer than k(L) = k(L) .

By lemma 1, there is an ultrafilter U L such that R ^ k(U) . By XQ , its image /(ti) converges to some point y in a w-compact set K f(L) . If 1/ is the filter based on U fl L , then K belongs to /(V) , a filter converging to y under w by X^ . Further, 0(U) 3 0 (A/) and 9 (M) — ►f(M) under u>* , for all M in U flL . This means that 0(U) — *- y under w* aj({A}) and hence under w , by

72 .

But k(U) is clearly finer than 0(U) x /(U) , a filter belonging to ft since both 0(U) and /(U) converge to y under w . In short, the still finer filter R also belongs to ft , as desired.

On the other hand, suppose that 0 —> / under L : ft , and that

(X3) each f(Q is m-compact

(X3) ft is a uniform convergence structure, and

o) is (V F,.iand Choquet.

92 Lemma 7. Then Q — *■ f under L : to* .

Proof. Again take L in L , let R be an ultrafilter finer than h(L) , and "choose" an ultrafilter il L such that R ~> h(U) . By , some point y of f(L) is the to-limit of f(U) . Consequently f(U) * f(U) belongs to fl , as do k(L) and k(U) . Further h(U) x /(U) = k(U) o {/(U) x f(U)} , as can be seen by comparing basis elements:

T(U) X /([/) = (T x { / } ) ( zo y (f(U) x f(U)) .

Thus h(U) and R both converge to y under to . In short, every ultrafilter finer than f(L) converges to some point of f(L) .

For each y in f(L) let be the set of all ultrafilters finer than h{L) which w-converge to y , and G be their intersection. * y Though G^ = 0 if Zy is void, in all cases G^ — y under nw . Further, as h(L) = fl{G : y t /(£)) , it converges to f(L) under u* , y as desired, provided that each G^ — * y under to .

So, suppose G is proper, and take an ultrafilter R finer than y G . Being finer than h(L) , it to-converges to some point x . Thus y y — >x under to (because to is j) and so R — *-y under to (because w is symmetric). Hence — *■ y under to (because to is Choquet).

Altogether, conditions to are equivalent to

(K) fl is a Choquet uniform convergence structure whose convergence to is ^2 1 anc* stron8ly compact-diagonal.

One should ask whether V forces fl to be principal, and hence generated by a classical uniformity. Consider the following example.

93 Let Q be the set of all complex numbers in the open right half plane, together with the origin. For each integer n > 0 , set

Qn = {x + iy € Q : \y\ < nx) ,

and let t and T be the usual topology and uniformity on Q .

To define the uniform rule fl , put U in fl iff U ? T and Q^2 £ U for some n . Clearly a filter w-converges to a non-zero limit iff it x-converges to that limit, while H — *•0 under u> iff H — *■ 0 under t and Q G H , for some n . (In more technical language,

By construction, w 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). Moreover, fl is a Choquet uniform convergence structure, but clearly not principal. In all, ft and ui satisfy V without being "classical".

These results may make better sense when one sees that X^ , X2 and X 3 help define a convergence on P . To be precise, consider a conver gence y °n Q , and a collection L of subsets of P satisfying , and let

C = {/ € QP : X2 and *3 hold} .

Clearly C contains all the constant maps.

The convergence B on P is defined as follows: F —> x under B iff /(F) — ►f(x) under y for all f in C , and L meets F fl x . By X^ and X 3 , each L is 6-compact and L generates 8 . Also, every member of C is B-y-continuous. (A function is said to be continuous if it "preserves convergence", that is, H — ►y implies 0 (H) — *g(y) •)

94 The converse is true too, at least if y is Choquet. To prove this, suppose g is B-y-continuous, g(T) — * y under y , and F meets L . Also, let G be the filter based on F flL .

Given an ultrafilter U o g(G) , one can "find" an ultrafilter V F such that g{\J flL) c (j . Choose a 8-compact set L in (/ flL . Then 1/ converges to some point x in L and g(V) — ►g(x ) under y . But since g(x) = y and the filter W based on V flL converges to x under B , the filter U converges to y under y . Hence g(G) — *-y under y , because y is Choquet. In all, g satisfies X2 and * 3 .

A global comparison of L : fl and L : id* is now easily made.

Theorem 8. Let L satisfy , ft satisfy V s and C be the set of functions satisfying X2 and X3 . Then the convergences L : ft and L : w* coincide on C .

Second Comparisons

Next let a and y be convergences on P and Q respectively, and let Con(a,y) stand for the set of all a-y-continuous functions. Despite its name, continuous convergence con(a,y) can be defined on , not just on Con(a,y): 0 — *■ f under con(a,y) iff 0(F) — *■ f(x) under y whenever F —x >■ under a . However as f converges continuously to f iff f is continuous, con(a,y) does not satisfy C j outside Con(a,y) . For more details, see [3], [5] or [l].

Theorem 9. Suppose that L generates a and that y is strongly diagonal. Then L : y* is finer than continuous convergence, on Con(a,y) .

Proof: Let 0 — *■ f under L : y* , and F converge to x under a . If G is the filter based on F fl L , then /(G) -*■ /(x) under a ,

95 while 0(F) ^ 0(G) and 0(C) —> f(G) under y* , for all G in F flL . Hence 0(F) — *•f(x) under y ’-y = y , as desired.

From the proof, one can see that this remains true if y is merely strongly compact-diagonal, but the members of L are cx-compact. In the uniform case, the comparison below is well known.

Theorem 10. Let ft be a uniform convergence structure on Q , and L cover a . Then L : ft is finer than continuous convergence, on Con{a,uj) .

Next, consider {^}-convergence, for any a-compact set K . 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 R1 1, symmetric and Choquet. P Then continuous convergence is finer than {K}:y*, on Q .

Lemma 12. Let K be a-compact, and SI be a Choquet uniform rule. Then continuous convergence is finer than (if) : ft , on Con (a, to) .

Theorem 13. Let y be Rj l J symmetric, Choquet and strongly compact- diagonal, SI be a Choquet uniform convergence structures and K the set of all a-compact sets.

(i) If a is compactly generated then K : y* coincides with continuous convergence, on Con(a,y) .

(ii) If a is locally compact then K : ft coincides with continuous convergences on Con(a,w) .

Theorem 14. If a is compactly generated and ft satisfies Vs then continuous convergence, K : ft and K : ai* all coincide, on Con (a, to) .

More generally, if K is a family of a-compact sets, satisfying , then a(K) is a compactly generated convergence finer than a .

96 Thus if U satisfies V , then both K : ft and K : coincide with con(a(K) ,u>) , on Con(a(K) ,w) and in particular, on 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. An obvious technical one concerns V : can R^ j be weakened, say, to R^ (I guess not) or strong compact- diagonal ity to compact-diagonality (I don't see how)?

Since the "constant function" map embeds y in con(a,y) , continuous convergence can be topological only if 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 a is locally compact or compactly generated. Similarly, if con(a,y) coincides with L-convergence, then a is "often" compactly generated. The problem is: what conditions on a and y 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.

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