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.


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)


Zbl 0522.54015
Full Text: DOI arXiv


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