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On a nice embedding and the Ascoli theorem. (English) Zbl 1058.54007
The author derives Ascoli-like theorems for some set-open topologies and gives Ascoli-like statements, wich are focused on the set of functions. The following notations are used: \({\mathcal F}(X)\), \({\mathcal F}_0(X)\), \({\mathcal F}(\varphi)\) and \({\mathcal F}_0(\varphi)\) are the families of all filters on \(X\), all ultrafilters on \(X\), all refining filters of a filter \(\varphi\) and all refining ultrafilters of \(\varphi\), respectively. \({\mathcal P} (X)\) denotes the power set of the set \(X\) and \({\mathcal P}_0(X)={\mathcal P} (X)\setminus\{\varphi\}\). For a point \(x\in X\), \(\dot x\) denotes the singleton filter generated from the base \(\{\{x\}\}\). For a topological space \((X,\tau)\), \(q_\tau\) denotes the convergence on \(X\) induced by \(\tau\), i.e., the relation \[ q:=\bigl\{(\varphi,x) \in{\mathcal F}(X)\times X: \varphi\supset\dot x\cap\tau\bigr\} \] between the filters on \(X\) and the points of \(X\). For \(B\in{\mathcal P}(X)\) and \({\mathcal A}\subset {\mathcal P} (X)\), set \(B^{-{\mathcal A}}= \{A\in{\mathcal A}:A\cap B\neq\varphi\}\) and \(B^{+ {\mathcal A}}= \{A\in{\mathcal A}:A\cap B=\varphi\}\), and \(\tau_{\ell,{\mathcal A} }\) is the topology on \({\mathcal A}\) generated by the subbase of all \(G^{-{\mathcal A}}\), \(G\in\tau\). For \(\varphi\neq \alpha\subset {\mathcal P}(X)\), \(\tau_{d,{\mathcal A}}\) is the topology on \({\mathcal A}\) which is generated from the subbase of all \(B^{+{\mathcal A}}\), \(B\in\alpha\) and \(G^{-{\mathcal A}}\), \(G\in\tau\).
Definition 2.4. Let \((X,\tau)\) be a topological space. A subset \(A\subset X\) is called weak relative complete in \(X\), iff \[ \forall\varphi \in{\mathcal F}(A)\cap q^{-1}_\tau(X): {\mathcal F}(\varphi)\cap q^{-1}_\tau(A) \neq\varphi, \] i.e., every filter on \(A\), which converges in \(X\), has a refinement, converging in \(A\).
Relating notions and the definition above the author gives the following statement:
Theorem 2.5. Let \((X,\tau)\) be a topological space, and let \(\alpha\subset{\mathcal P}(X)\) consist of weak relative complete subsets of \(X\). Then it holds for any \({\mathcal A}\) with \(Cl (X)\subset {\mathcal A}\subset {\mathcal P}(X): ({\mathcal A}_0,\tau_{\alpha,{\mathcal A}_0})\) is compact \(\Leftrightarrow(X,\tau)\) is compact, where \(Cl(X)\) denotes the family of all closed subsets of \(X\), and \({\mathcal A}_0={\mathcal A} \setminus \{\varphi\}\).
Relating Mizokami’s result that for Hausdorff spaces \(X\) and \(Y\) the function space \(C(X,Y)\) endowed with the compact-open topology is isomorphic to a closed subspace of the function space \(C(K(X),K(Y))\), endowed with the pointwise topology, where \(K (X), K(Y)\) are the families of all compact subsets of \(X\), \(Y\), respectively, equipped with Vietoris topology, the author gives the following:
Lemma 3.4. Let \((X,\tau)\), \((Y,\sigma)\) be topological spaces, let \({\mathcal A}\subset{\mathcal P}_0(X)\) contain the singletons and let \({\mathcal H} \subset Y^X\) be endowed with \(\tau_{\mathcal A}\). Then the map \[ \mu: {\mathcal H}\to\mu ({\mathcal H})=\bigl\{\mu(f): \mu(f):A\to F(A), f\in {\mathcal H}\bigr\} \subset{\mathcal P}_0(Y)^{\mathcal A} \] is open, continuous and bijective, where \({\mathcal A}\) and \({\mathcal P}_0(Y)\) are equipped with Vietoris topology, and \({\mathcal P}_0(Y)^{\mathcal A}\) with pointwise topology.
Definition 3.6. Let \((X,\tau)\), \((Y,\sigma)\) be topological spaces and \({\mathcal A}\subset{\mathcal P}_0(X)\). A subset \({\mathcal H}\subset Y^X\) is called \({\mathcal A}\)-evenly continuous, iff for all \(A\in{\mathcal A}\) holds \[ \forall {\mathcal F}\in{\mathcal F}_0 ({\mathcal H}),\;\varphi\in {\mathcal F}(A),x\in X:\bigl({\mathcal F} (x)@>\sigma>> y\bigr)\wedge(\varphi @>\tau>> x)\Rightarrow {\mathcal F} (\varphi) @>\sigma>>y. \] \({\mathcal H}\) is called evenly continuous, iff it is \(\{X\}\)-evenly continuous. \({\mathcal H}\) is called evenly continuous on \(K\subset X\), iff \({\mathcal H}_\mathbb{K}=\{f|_K: \mathbb{K}\to Y: f\in{\mathcal H}\}\) is evenly continuous.
The following propositions are shown to be related to the Ascoli theorem: Lemma 3.10. (Essential Ascoli) Let \((X,\tau)\), \((Y,\sigma)\) be topological spaces and \({\mathcal A}\subset{\mathcal P}_0 (X)\). Let \({\mathcal H}\subset C(X,Y)\) and \({\mathcal F}\) an ultrafilter on \({\mathcal H}\), which converges pointwise to a function \(g\in C(X,Y)\). Then the following holds:
1. If \({\mathcal A}\) consists only of relative compact subsets of \(X\), \({\mathcal H}\) is \({\mathcal A}\)-evenly continuous and the images of all members of \({\mathcal A}\) under \(g\) are closed in \(Y\), then \(\mu({\mathcal F})\) converges pointwise to \(\mu(g)\) in \(C({\mathcal A},C_Y({\mathcal A}))\), where \(C_Y({\mathcal A})= \{f(A):A\in{\mathcal A}\), \(f\in C(X,Y)\}\).
2. If \({\mathcal A}\) consists only of compact subsets of \(X\) and \({\mathcal H}\) is evenly continuous on all members of \({\mathcal A}\), then \(\mu({\mathcal F})\) converges pointwise to \(\mu(g)\) on \(C({\mathcal A}, C_Y({\mathcal A}))\).
Theorem 3.14. Let \((X,\tau)\), \((Y,\sigma)\) be topological spaces and \({\mathcal A}\subset{\mathcal P}_0(X)\) contain singletons. Then a set \({\mathcal H}\subset Y^X\) is relative compact in \((Y^X,\tau_a)\) if and only if
1. for all ultrafilters \({\mathcal F}\) on \({\mathcal H}\) with \({\mathcal F} @>p>> f\in Y^X\) exists a function \(g\in Y^X\) such that \(\mu({\mathcal F}) @>p>> \mu(g) \in{\mathcal P}_0 (Y)^{\mathcal A}\), where \({\mathcal P}_0(Y)\) is equipped with the Vietoris topology, and
2. for all \(A\), the family \(\mu({\mathcal H}(A)= \{f(A): f\in{\mathcal H}\}\) is relative compact in \({\mathcal P}_0(Y)\) with respect to the Vietoris topology.
54C35 Function spaces in general topology
54D30 Compactness
54C05 Continuous maps
54C25 Embedding
54C60 Set-valued maps in general topology
54B20 Hyperspaces in general topology
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
46E10 Topological linear spaces of continuous, differentiable or analytic functions