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Group-valued continuous functions with the topology of pointwise convergence. (English) Zbl 1195.54040

Let \(X\), \(Y\) be Tychonoff spaces, \(G\) a topological group with identity \(e\), and let \(C_p(X, Y)\) denote the group of all continuous mappings from \(X\) into \(G\) with the topology of pointwise convergence. We say that \(X\) and \(Y\) are \(G\)-equivalent in case \(C_p(X, G)= C_p(Y, G)\). The object of this paper is to study in detail the topological properties of \(X\) which are preserved under \(G\)-equivalence for various choices of \(G\). We say that \(X\) is \(G\)-regular in case, for each closed set \(F\subset X\) and each point \(p\in X\setminus F\), there exist \(f\in C_p(X, G)\) and \(g\in G\) \((g\neq e)\) such that \(f(p)= g\) and \(f(F)= \{e\}\). The authors show, among other things, that if \(G\) is an NSS-group (no small subgroups), and \(X\) is \(G\)-regular, then \(G\)-equivalence preserves the properties (a) pseudo compactness and (b) compact metrizability. Part (a) generalizes a result of A. V. Arkhangel’skij [Sov. Math., Dokl. 25, 852–855 (1982); translation from Dokl. Akad. Nauk SSSR 264, 1289–1292 (1982; Zbl 0522.54015)], where \(G= R\) is the real additive group. Another result says that \(T\)-equivalence implies \(G\)-equivalence for every precompact Abelian group \(G\), where \(T\) is the circle group. Various counterexamples and open questions are presented.

MSC:

54C35 Function spaces in general topology
22A05 Structure of general topological groups
46E10 Topological linear spaces of continuous, differentiable or analytic functions
54H11 Topological groups (topological aspects)

Citations:

Zbl 0522.54015
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References:

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