STRONG α-FAVORABILITY OF THE

(GENERALIZED) COMPACT-OPEN TOPOLOGY

Peter J. Nyikos and László Zsilinszky

Abstract. Strong α-favorability of the compact-open topology on the space of continuous functions, as well as of the generalized compact-open topology on continuous partial functions with closed domains is studied.

1. Introduction

Spaces of partial maps have been studied for various applications throughout the century ([Ku1-2], [AB], [BB], [Ba], [DN1-2], [Fi], [KS], [La], [Se], [St], [Wh], [Za]). In particular, the so-called generalized compact-open topology on the space of continuous partial functions with closed domains proved to be a useful tool in mathematical economics ([Ba]), in convergence of dynamic programming models ([La], [Wh]) or more recently in the theory of differential equations ([BC]). This topology was also scrutinized from purely topological point of view e.g. in [BCH], [Ho], [HZ1-2], where among others, separation axioms and some completeness properties (such as Baireness, weak α-favorability, Čech-completeness, complete metrizability) of the generalized compact-open topology have been investigated.

Our paper continues in this research by looking at strong α-favorability in this setting. Section 3 contains our results on strong α-favorability of τ_{C} as well as a short proof of a recent theorem of Holá on complete metrizability of τ_{C}.

We will rely on the close connection that exists between the generalized compactopen topology, the ordinary compact-open topology τ_{CO} ([MN1]) and the Fell topology τ_{F} on the hyperspace of nonempty closed subsets of a topological space ([Be], [KT]). This connection and some other auxiliary material is described at the end of Section 1, while in Section 2 we list results about strong α-favorability of τ_{CO} and τ_{F}, respectively, needed for proving our main results; a generalization of a theorem of Ma on weak α-favorability of the compact-open topology is also given.

Let X and Y be Hausdorff spaces. Denote by CL(X) the family of nonempty closed subsets of X and by K(X) the nonempty compact subsets of X. For any B ∈ CL(X) and a topological space Y, C(B, Y) will stand for the space of all continuous functions from B to Y. A partial map is a pair (B, f) such that B ∈ CL(X) and f ∈ C(B, Y). Denote by P = P(X, Y) the family of all partial maps.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 54C35; 54B20, 90D44.

Key words and phrases. Generalized compact-open topology, Fell topology, compact-open topology, strong α-favorability, weak α-favorability.

Define the so-called generalized compact-open topology τ_{C} on P as the topology having subbase elements of the form

[U] = {(B, f) ∈ P : B ∩ U ≠ ∅},
[K : I] = {(B, f) ∈ P : f(K ∩ B) ⊂ I},

where U is open in X, K ⊂ X is compact and I is an open (possibly empty) subset of Y. We can assume that the I’s are members of some fixed open base for Y.

The compact-open topology τ_{CO} on C(X, Y) has subbase elements of the form

[K, I] = C(X, Y) ∩ [K : I] = {f ∈ C(X, Y) : f(K) ⊂ I},

where K ⊂ X is compact and I ⊂ Y is open; C_{k}(X) (see [MN1]) stands for (C(X, Y), τ_{CO}) with Y = ℝ (the reals). Note, that C_{k}(X) is a topological group, so a typical basic open neighborhood of f ∈ C_{k}(X) is of the form f +[K, I] = {f +f′ ∈ C_{k}(X) : f′ ∈ [K, I]}, where K ∈ K(X) and I is a bounded open neighborhood of zero. We will also use that, if X is a Tychonoff space, then f + [K, I] ⊂ f′ + [K′, I′] implies K ⊃ K′.

Denote by τ_{F} the so-called Fell topology on CL(X) having subbase elements of the form {A ∈ CL(X) : A ∩ V ≠ ∅} with V open in X, plus sets of the form {A ∈ CL(X) : A ⊂ V } with V co-compact in X. For notions not defined in the paper see [En].

In the strong Choquet game (cf. [Ch] or [Ke]) two players, α and β, take turns in choosing objects in the topological space X with an open base B: β starts by picking (x_{0}, V_{0}) from E(X) = E(X, B) = {(x, V) ∈ X × B : x ∈ V } and α responds by U_{0} ∈ B with x_{0} ∈ U_{0} ⊂ V_{0}. The next choice of β is some couple (x_{1}, V_{1}) ∈ E(X, B) with V_{1} ⊂ U_{0} and again α picks U_{1} with x_{1} ∈ U_{1} ⊂ V_{1} etc. Player α wins the run (x_{0}, V_{0}), U_{0}, …, (x_{n}, V_{n}), U_{n}, … provided ⋂_{n} U_{n} = ⋂_{n} V_{n} ≠ ∅, otherwise β wins. A winning tactic for α (cf. [Ch]) is a function σ : E(X, B) → B such that α wins every run of the game compatible with σ, i.e. such that U_{n} = σ(x_{n}, V_{n}) for all n. The strong Choquet game is α-favorable if α possesses a winning tactic; in this case X is called strongly α-favorable (or a strong Choquet space - cf. [Ke]). We will need the following facts about the strong Choquet game:

Proposition 1.1.

(i) Let X be metrizable. Then X is completely metrizable if and only if X is strongly α-favorable.

(ii) If X is locally compact, then X is strongly α-favorable.

(iii) Let f : X → Y be continuous, open and onto. If X is strongly α-favorable,

so is Y. (iv) The product of any collection of strongly α-favorable spaces is strongly α-

favorable.

Proof. It is not hard to show that (ii)-(iv) holds (cf. [Ke], Exercise 8.16); as for (i), see [Ch], Theorem 8.7 or [Ke], Theorem 8.17. □

STRONG α-FAVORABILITY OF FUNCTION SPACES 3

The Banach-Mazur game (see [HM] or the Choquet game in [Ke]) is played as the strong Choquet game except that β’s choice is just a nonempty open set contained in the previous move of α. A space X is called weakly α-favorable if α possesses a winning strategy in the Banach-Mazur game (i.e. a function defined on nests of nonempty open sets of odd length picking for α the set that wins the Banach-Mazur game for α no matter what β chooses). Note that β has no winning strategy in the Banach-Mazur game if and only if X is a Baire space (i.e. countable intersections of dense open sets are dense - cf. [Ke] or [HM]), consequently, weakly α-favorable spaces are Baire spaces.

The restriction mapping

η : (CL(X), τ_{F}) × (C(X, Y), τ_{CO}) → (P, τ_{C})

is defined as η((B, f)) = (B, f ↾_{B}).

Clearly, η is onto provided continuous partial functions with closed domain are continuously extendable over X. We can say more about η if we assume that X, Y have property (P), i.e. if X, Y are such that partial continuous functions with closed domains are continuously extendable over X and there exists an open base V for Y closed under finite intersections such that for each nonempty K ∈ K(X) and V ∈ V, every function f ∈ C(K, V) is extendable to some f^{∗} ∈ C(X, V). A fundamental result about η is as follows (see [HZ1], Section 3):

Proposition 1.2.

(i) If X, Y have property (P), then η is open, continuous and onto.

(ii) If X is paracompact and Y is locally convex completely metrizable or if X is T_{4} and Y ⊂ ℝ is an interval, then X, Y have property (P). In particular, η is open, continuous and onto in this case.

2. Strong α-favorability of τ_{CO} and τ_{F}

As for strong α-favorability of the Fell topology, we have:

Theorem 2.1.

(i) If X is locally compact, then (CL(X), τ_{F}) is locally compact (and hence strongly α-favorable).

(ii) If X is a strongly α-favorable space such that the countable subsets of X are closed, then (CL(X), τ_{F}) is strongly α-favorable.

Proof. (i) See [Be], Corollary 5.1.4.

(ii) In our case K(X) is a weakly Urysohn family, i.e. if S ∈ K(X) and A ⊂ S^{c}, then there exists T ∈ K(X) with A ⊂ T^{c} ⊂ S^{c} such that \overline{E} ⊂ S^{c} for all countable E ⊂ T^{c} (we can choose T = S). Consequently, Theorem 5.1 of [Zs] yields the desired result. □

Recall that a Hausdorff space X is hemicompact ([En], Excercise 3.4.E) provided in the family of all compact subspaces of X ordered by inclusion there exists a countable cofinal subfamily.

Theorem 2.2. If X is locally compact paracompact and Y is completely metrizable, then (C(X, Y), τ_{CO}) is strongly α-favorable.

Proof. The proof of Theorem 5.3.1 in [MN1] can be modified to get the result: write X = ⨁_{t∈T} X_{t}, where each X_{t} is locally compact and hemicompact ([En], Theorem 5.1.27). Then (C(X_{t}, Y), τ_{CO}) is completely metrizable for all t ∈ T ([MN1], Exercise 5.8.1(a)). Therefore, (C(X, Y), τ_{CO}) is homeomorphic to the product ∏_{t∈T} (C(X_{t}, Y), τ_{CO}) ([MN1], Corollary 2.4.7) of completely metrizable spaces, hence, in view of Proposition 1.1 (i) and (iii), (C(X, Y), τ_{CO}) is strongly α-favorable. □

A space X is a q-space if for each x ∈ X there is a sequence {G_{n}}_{n∈ω} of open neighborhoods of x such that whenever x_{n} ∈ G_{n} for all n, the set {x_{n}}_{n∈ω} has a cluster point. Notice that 1st countable or locally compact (even Čech-complete) spaces are q-spaces. The next result generalizes Theorem 1.2 of [Ma] about weak α-favorability of the compact-open topology (see also [MN2]):

Theorem 2.3. Let X be a q-space. Then the following are equivalent:

(i) C_{k}(X) is strongly α-favorable; (ii) C_{k}(X) is weakly α-favorable; (iii) X is locally compact and paracompact.

Proof. (i)⇒(ii) Clear.

(ii)⇒(iii) X is locally compact by Theorem 4.4 of [MN2], since weakly α-favorable spaces are Baire spaces. Paracompactness of X follows from Theorem 1.2 of [Ma].

(iii)⇒(i) See Theorem 2.2 □

Proposition 2.4. Let X be a T_{4} space with the countable subsets closed and discrete. Then C_{k}(X) is strongly α-favorable.

Proof. Let (f, U) ∈ E(C_{k}(X)) with U = f + [K, I] and diam(I) < ∞ (the diameter of I). Define σ(f, U) = f + [K, J], where J is an open neighbourhood of zero such that diam(J) = \frac{1}{2}diam(I). Then σ is a winning strategy for α: let (f_{0}, U_{0}), V_{0}, …, (f_{n}, U_{n}), V_{n}, … be a run of the strong Choquet game in C_{k}(X), where

U_{n} = f_{n} + [K_{n}, I_{n}], V_{n} = σ(f_{n}, U_{n}),

K_{n} ∈ K(X) and I_{n} is an open neighborhood of zero (n ∈ ω). Then U_{n+1} ⊂ V_{n} ⊂ U_{n} for each n ∈ ω, so K_{n+1} ⊃ K_{n} and diam(I_{n+1}) ≤ \frac{1}{2}diam(I_{n}); consequently, for each x ∈ K = ⋃_{n∈ω} K_{n}, the sequence {f_{n}(x)}_{n∈ω} converges to some f(x) ∈ ℝ. Observe that in our case the K_{n}’s are finite and hence K is closed and discrete, so the function f : K → ℝ defined above is continuous. If we extend f to some f^{∗} ∈ C(X, ℝ), we have f^{∗} ∈ ⋂_{n∈ω} U_{n} and α wins the run. □

3. Strong α-favorability of τ_{C}

Theorem 3.1. Assume that X, Y have property (P). If both (C(X, Y), τ_{CO}) and (CL(X), τ_{F}) are strongly α-favorable, so is (P, τ_{C}).

Proof. The restriction mapping is continuous, open and onto by Proposition 1.2(i), so Proposition 1.1(iv) and (iii) applies. □

The next theorem generalizes Corollary 4.4(i) of [HZ1]:

Theorem 3.2. Let X be a locally compact, paracompact space and Y a locally convex completely metrizable space. Then (P, τ_{C}) is strongly α-favorable.

Proof. (CL(X), τ_{F}) and (C(X, Y), τ_{CO}) are strongly α-favorable by Theorem 2.1(i) and Theorem 2.2, so Proposition 1.2 and Theorem 3.1 yields the desired result. □

As a corollary of Theorem 3.2 we get the following theorem of Holá ([Ho], Theorem 3.3):

Theorem 3.3. Let X be a Tychonoff space and Y a locally convex completely metrizable space. Then the following are equivalent:

(i) (P, τ_{C}) is completely metrizable; (ii) X is a locally compact 2nd countable space.

Proof. In view of [Ho] (Theorem 2.4), (P, τ_{C}) is metrizable if and only if X is locally compact and 2nd countable, so the implication (i)⇒(ii) immediately follows. As for (ii)⇒(i), use Theorem 3.2 and Proposition 1.1(i). □

We will now study strong α-favorability of τ_{C} for Y = ℝ to show that Theorem 3.2 is not reversible, i.e. that local compactness plus paracompactness is not necessary for strong α-favorability of the generalized compact-open topology.

Theorem 3.4. Let Y = ℝ and X be a T_{4} strongly α-favorable space with the countable subsets closed discrete. Then (P, τ_{C}) is strongly α-favorable.

Proof. (CL(X), τ_{F}) and C_{k}(X) are strongly α-favorable by Theorem 2.1(ii) and Theorem 2.4, respectively, hence Proposition 1.2(ii) and Theorem 3.1 applies. □

To demonstrate that Theorem 3.2 is not reversible we need (by Theorem 3.4) the following:

Example 3.5. There exists a T_{4} non-paracompact, strongly α-favorable space with the countable subsets closed discrete.

Proof. The space with the required properties is X = {x ∈ ω_{2} : cf x > ω}, which is a stationary subset of ω_{2} and hence X is T_{4} and, by the Pressing Down Lemma, not paracompact. Further, by the definition of X, no countable subset of X clusters, thus, countable subsets of X are closed and discrete. To show that X is strongly α-favorable, put σ(x, U) = U for every (x, U) ∈ E(X) with U = (a, x] ∩ X.

Then σ is a winning tactic for α, since if (x_{0}, U_{0}), U_{0}, …, (x_{n}, U_{n}), U_{n}, … is a run of the strong Choquet game compatible with σ, then there exists some n_{0} ∈ ω with x = x_{n} = x_{m} for all m, n ≥ n_{0} (otherwise {x_{n}}_{n} would have a subsequence of order type ω^{∗}), whence x ∈ ⋂_{n∈ω} U_{n}. □

References

[AB] A. Abd-Allah and R. Brown, A compact-open topology on partial maps with open domains, J. London Math. Soc. 21 (1980), 480-486.

[Ba] J. Back, Concepts of similarity for utility functions, Journal of Math. Economics 1 (1976), 721-727.

[BB] P. I. Booth and R. Brown, Spaces of partial maps, fibred mapping spaces and the compact open topology,, Topology Appl. 8 (1978), 181-195.

[BC] P. Brandi and R. Ceppitelli, Existence, uniqueness and continuous dependence for hereditary differential equations, J. Diff. Equations 81 (1989), 317 - 339.

[BCH] P. Brandi, R. Ceppitelli and L’. Holá, Topological properties of a new graph topology, Jour. Convex. Anal. 5 (1998), 1-12.

[Be] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht, 1993.

[Ch] G. Choquet, Lectures on Analysis I., Benjamin, New York, 1969.

[DN1] A. Di Concilio and S.A. Naimpally, Proximal set-open topologies and partial maps, Boll. U.M.I., to appear.

[DN2] A. Di Concilio and S.A. Naimpally, Function space topologies on (partial) maps, Recent progress in function spaces, Quaderni di Matematica (G. Di Maio and L’. Holá, eds.), vol. 3, 1999.

[En] R. Engelking, General Topology, Helderman, Berlin, 1989.

[Fi] V.V. Filippov, Basic topological structures of the theory of ordinary differential equations, Topology in Nonlinear Analysis, Banach Centrum Publications 35 (1996), 171-192.

[HM] R.C. Haworth and R.A. McCoy, Baire spaces, Dissertationes Math. 141 (1977), 1-73.

[Ho] L’. Holá, Complete metrizability of generalized compact-open topology, Topology Appl. 91 (1999), 159-167.

[HZ1] L’. Holá and L. Zsilinszky, Completeness properties of the generalized compact-open topology on partial functions with closed domains, Topology Appl. 110 (2001), 303-321.

[HZ2] L’. Holá and L. Zsilinszky, Čech-completeness and related properties of the generalized compact-open topology, preprint.

[Ke] A.S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1994.

[KS] H. P. Künzi and L. B. Shapiro, On simultaneous extension of continuous partial functions, Proc. Amer. Math. Soc. 125 (1997), 1853-1859.

[KT] E. Klein and A. Thompson, Theory of Correspondences, Wiley, New York, 1975.

[Ku1] K. Kuratowski, Sur l’espace des fonctions partielles, Ann. Mat. Pura Appl. 40 (1955), 61-67.

[Ku2] K. Kuratowski, Sur une méthode de métrisation complète de certains espaces d’ensembles compacts, Fund. Math. 43 (1956), 114-138.

[La] H.J. Langen, Convergence of dynamic programming models, Mathematics of Operations Research 6 (1981), 493-512.

[Ma] D. K. Ma, The Cantor tree, the γ-property and Baire function spaces, Proc. Amer. Math. Soc. 119 (1993), 903-913.

[MN1] R.A. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Springer-Verlag, Berlin, 1988.

[MN2] R.A. McCoy and I. Ntantu, Completeness properties of function spaces, Topology Appl. 22 (1986), 191-206.

[Se] G. R. Sell, On the Fundamental Theory of Ordinary Differential Equations, J. Differential Equations 1 (1965), 3710-392.

[St] E.N. Stepanova, Extension of continuous functions and metrizability of paracompact pspaces, Mat. Zametki 53 (1993), 92-101; English transl., Math. Notes 53 (1993), 308-314.

[Wh] W. Whitt, Continuity of Markov Processes and Dynamic Programs, Yale University, 1975.

[Za] S.K. Zaremba, Sur certaines familles de courbes en relation avec la theorie des equations differentielles, Rocznik Polskiego Tow. Matemat. 15 (1936), 83-105.

[Zs] L. Zsilinszky, Topological Games and Hyperspace Topologies, Set-Valued Anal. 6 (1998), 187-207.

Department of mathematics, University of South Carolina, Columbia, SC 29208

   E-mail address: nyikos@math.sc.edu

Department of Mathematics and Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372

   E-mail address: laszlo@nat.uncp.edu