Group-valued continuous functions with the topology of pointwise convergence

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

2009-07-28

arxiv.org/abs/0907.4941v2

Topology and its Applications, 157 (2010), 1518-1540

Mathematics

General Topology

Two references were added and one reference was updated. Question 11.1 was resolved in arXiv:0909.2381 [math.GN]. The bibliogr

Scientific paper

10.1016/j.topol.2009.06.022

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.

Affiliated with

Also associated with

No associations

Rating

Group-valued continuous functions with the topology of pointwise convergence does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Group-valued continuous functions with the topology of pointwise convergence, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Group-valued continuous functions with the topology of pointwise convergence will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-682346