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Topological group criterion for \(C(X)\) in compact-open-like topologies. I. (English) Zbl 1166.54007

Given a space \(Y\), denote by \({\mathcal K}(Y)\) the family of all compact subsets of \(Y\); for any \(K\in {\mathcal K}(Y)\) let \(O(K)=\{f\in C(Y): f(K)=\{0\}\}\). The family \({\mathcal A}=\{O(K): K\in {\mathcal K}(Y)\}\) generates a topology \(\tau\) on \(C(Y)\) such that \({\mathcal A}\) is a local base of \((C(Y),\tau)\) at the origin. This topology is called the compact-zero topology on \(C(Y)\).
For a compact space \(X\) consider a filter base \({\mathcal F}\) of dense open \(F_\sigma\)-subsets of \(X\) and let \({\mathcal F}_\delta\) be the family of all countable intersections of elements of \({\mathcal F}\). Any set \(F\in {\mathcal F}_\delta\) is dense in \(X\) so the restriction map \(\pi_F: C(X)\to C(F)\) defined by \(\pi(f)=f|F\) for any \(f\in C(X)\), is injective. Let \(co(F)\) be the compact-open topology on \(C(F)\) and denote by \(cz(F)\) the compact-zero topology on the set \(C(F)\).
For every set \(F\in {\mathcal F}_\delta\) the families \(\tau_F=\{\pi^{-1}(U) : Y\in co(F)\}\) and \(\sigma_F=\{\pi^{-1}(V) : V\in cz(F)\}\) are topologies on \(C(X)\) and hence \(\tau_{\mathcal F}= \bigwedge\{ \tau_F : F\in {\mathcal F}_\delta\} \) and \(\sigma_{\mathcal F}= \bigwedge\{ \sigma_F : F\in {\mathcal F}_\delta\} \) are also topologies on \(C(X)\).
The authors study the situation when \((C(X),\tau)\) is a topological group for some (or each) \(\tau\in \{\tau_{\mathcal F}, \sigma_{\mathcal F}\}\). They basically consider the case when \(X=\beta Y\) and \({\mathcal F}\) is the family \({\mathcal C}\) of all cozero subsets of \(X\) which contain \(Y\).
It is established, among other things, that if \(X=\beta Y\) for some space \(Y\) such that \(\upsilon Y\) is Lindelöf and Čech-complete then \(\tau_{\mathcal C}\) and \(\sigma_{\mathcal C}\) are group topologies on \(C(X)\). The authors also show that if \(Y\) is a discrete space of cardinality \(\omega_1\) and \(X= \beta Y\) then \(\tau_{\mathcal C}\) and \(\sigma_{\mathcal C}\) are not group topologies on \(C(X)\).

MSC:

54C35 Function spaces in general topology
18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
22A99 Topological and differentiable algebraic systems
46A40 Ordered topological linear spaces, vector lattices
03E05 Other combinatorial set theory
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