Group-valued Continuous Functions with the Topology of Pointwise Convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F ⊆ X and every point x ∈ X \ F , there exist f ∈ Cp(X,G) and g ∈ G \ {e} such that f(x) = g and f(F ) ⊆ {e}; (b) G-regular provided that there exists g ∈ G \ {e} such that, for each closed set F ⊆ X and every point x ∈ X \ F , one can find f ∈ Cp(X,G) with f(x) = g and f(F ) ⊆ {e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic. We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel’skĭı as a particular case (when G = R). We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G -regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP. We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed. In notation and terminology we follow [7] and [10] if not stated otherwise. All topological spaces are assumed to be Tychonoff (that is, completely regular T1 spaces), and all topological groups are assumed to be Hausdorff. By N we denote the set of all natural numbers, ω stays for the least nonzero limit ordinal, Z is the discrete additive group of integers, R is the additive group of reals with its usual topology, T stays for the quotient group R/Z, and Z(n) denotes the cyclic group of order n (with the discrete topology). The identity element of a group G is denoted by eG, or simply by e when there is no danger of confusion.