Atlas: Locally precompact groups: Connectedness and local connectedness by Gabor Lukacs

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Spring Topology and Dynamical Systems Conference 2008
March 13-15, 2008
University of Wisconsin Milwaukee and Marquette University
Milwaukee, WI, USA

Organizers
Ric Ancel, Karen Brucks, Craig Guilbault, Chris Hruska, Suzanne Hruska, Boris Okun (UWM); Paul Bankston (Marquette); Lois Kailhofer (Alverno College).

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Locally precompact groups: Connectedness and local connectedness
by
Gábor Lukács
University of Manitoba
Coauthors: W. W. Comfort

A subset B of a (Hausdorff) topological group G is said to be precompact (or bounded) if for every neighborhood U of the identity in G, there is a finite subset F ⊆ G such that B subseteq (FU) ∩(UF). Precompact groups have been a focus of interest since Comfort and Ross's paper of 1966, where the authors proved that a group G is pseudocompact if and only if it is precompact and G_d-dense in its completion [G\tilde] (cf. [3], [6], [1]).

A group G is locally precompact if it contains a precompact neighborhood of the identity. The completion of a locally precompact group is locally compact, and thus such groups are precisely the subgroups of locally compact groups. Comfort and Trigos-Arrieta extended the Comfort-Ross criterion, and proved that a locally precompact group G is locally pseudocompact if and only if it is G_d-dense in [G\tilde] (cf. [4]). Locally pseudocompact groups were also studied by Sanchis (cf. [6]).

Let G be a locally precompact group, and let L = [G\tilde] be its completion. In this talk, we are concerned with the relationship between connectedness properties of G and L. Let G₀ and L₀ denote the connected component of the identity in G and L, respectively. Clearly, one has

cl_L G₀ ⊆ cl_L (L₀ ∩G) ⊆ L₀,

but are any of these inclusions strict? Will these inclusions be strict under the additional assumption that G is (locally) pseudocompact? What does the intersection L₀ ∩G look like? In this talk, we present answers to these questions.

References

[1] A. V. Arhangel'skii. On a theorem of W. W. Comfort and K. A. Ross. Comment. Math. Univ. Carolin., 40(1):133-151, 1999.

[2] B. Banaschewski. Local connectedness of extension spaces. Canad. J. Math., 8:395-398, 1956.

[3] W. W. Comfort and Kenneth A. Ross. Pseudocompactness and uniform continuity in topological groups. Pacific J. Math., 16:483-496, 1966.

[4] W. W. Comfort and F. Javier Trigos-Arrieta. Locally pseudocompact topological groups. Topology Appl., 62(3):263-280, 1995.

[5] M. Henriksen and J. R. Isbell. Local connectedness in the Stone-Cech compactification. Illinois J. Math., 1:574-582, 1957.

[6] S. Hernández and M. Sanchis. G_d-open functionally bounded subsets in topological groups. Topology Appl., 53(3):289-299, 1993.

[7] Manuel Sanchis. Continuous functions on locally pseudocompact groups. Topology Appl., 86(1):5-23, 1998. Special issue on topological groups.

Date received: February 28, 2008