| 研究概要 |
We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The ``compact-like'' properties we consider include (local) compactness, (local) omega-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is a sample of our characterizations:
(i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups.
(ii) An abelian topological group group G is a Lie group if and only if G is locally minimal, locally precompact and all closed metric zero-dimensional subgroups of G are discrete.
(iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups.
(iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.
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