## 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)

Zbl 0522.54015
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### References:

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