Group-valued continuous functions with the topology of pointwise convergence

Abstract

Let G be a topological group with the identity element e. Given a space X, we denote by C p ( 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 ∈ C p ( 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 ∈ C p ( X , G ) with f ( x ) = g and f ( F ) ⊆ { e } . Spaces X and Y are G-equivalent provided that the topological groups C p ( X , G ) and C p ( 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 C p ( X , G ) . Since R -equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical C p -theory of Arhangel'skiĭ 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 C p ( X , G ) is TAP, and (ii) for a metrizable NSS group G, a G ⋆ -regular space X is compact if and only if C p ( X , G ) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if C p ( 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 C p ( 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.