# Group-valued continuous functions with the topology of pointwise
convergence

Research paper by **Dmitri Shakhmatov, Jan Spěvák**

Indexed on: **23 Apr '10**Published on: **23 Apr '10**Published in: **Mathematics - General Topology**

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#### Abstract

We denote by C_p(X,G) the group of all continuous functions from a space X to
a topological group G endowed with the topology of pointwise convergence. We
say that 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'skii
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). 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, sigma-compactness, the
property of being a Lindelof Sigma-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.