Locally Precompact Groups: (Local) Realcompactness and Connectedness

W. W. Comfort; G. Lukács

doi:10.5802/jolt.600

W. W. Comfort; G. Lukács

Journal of Lie Theory, Volume 20 (2010) no. 2, pp. 347-374

Abstract

A theorem of A. Weil asserts that a topological group embeds as a (dense) subgroup of a locally compact group if and only if it contains a non-empty precompact open set; such groups are called "locally precompact. Within the class of locally precompact groups, the authors classify those groups with the following topological properties:
(1) Dieudonné completeness; (2) local realcompactness; (3) realcompactness; (4) hereditary realcompactness; (5) connectedness; (6) local connectedness; (7) zero-dimensionality.
They also prove that an abelian locally precompact group occurs as the quasi-component of a topological group if and only if it is "precompactly generated", that is, it is generated algebraically by a precompact subset.

arXiv Zbl

DOI: 10.5802/jolt.600

Classification: 22A05, 54H11, 22B05, 22C05
Keywords: Precompact group, precompactly generated group, locally precompact group, Weil completion, Dieudonné complete group, locally Dieudonné complete group, realcompact group, locally realcompact group, connected group, locally connected group, omega-balanced g

@article{JOLT_2010_20_2_a6,
     author = {W. W. Comfort and G. Luk\'acs},
     title = {Locally {Precompact} {Groups:} {(Local)} {Realcompactness} and {Connectedness}},
     journal = {Journal of Lie Theory},
     pages = {347--374},
     year = {2010},
     volume = {20},
     number = {2},
     doi = {10.5802/jolt.600},
     zbl = {1201.22001},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.600/}
}

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W. W. Comfort; G. Lukács. Locally Precompact Groups: (Local) Realcompactness and Connectedness. Journal of Lie Theory, Volume 20 (2010) no. 2, pp. 347-374. doi: 10.5802/jolt.600

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