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Register a new userLocally precompact groups: (Local) realcompactness and connectedness
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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: Dieudonne completeness; local realcompactness; realcompactness; hereditary realcompactness; connectedness; local connectedness; 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.
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We discuss various modifications of separability, precompactnmess and narrowness in topological groups and test those modifications in the permutation groups $S(X)$ and $S_{<omega}(X)$.
An embeddability criterion for zero-dimensional metrizable topological spaces in zero-dimensional metrizable topological groups is given. A space which can be embedded as a closed subspace in a zero-dimensional metrizable group but is not strongly zero-dimensional is constructed; thereby, an example of a metrizable group with noncoinciding dimensions ind and dim is obtained. It is proved that one of Kuleszas zero-dimensional metrizable spaces cannot be embedded in a metrizable zero-dimensional group.
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In his seminal work cite{pal:61}, R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if $H$ is a compact subgroup of a locally compact group $G$ and $S$ is a small (in the sense of Palais) $H$-slice in a proper $G$-space, then the action map $Gtimes Sto G(S)$ is open. This is applied to prove that the slicing map $f_S:G(S)to G/H$ is continuos and open, which provides an external characterization of a slice. Also an equivariant extension theorem is proved for proper actions. As an application, we give a short proof of the compactness of the Banach-Mazur compacta.
We answer several questions of I.Protasov and E.Zelenyuk concerning topologies on groups determined by T-sequences. A special attention is paid to studying the operation of supremum of two group topologies.
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