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Compactly Generated
In mathematics, compactly generated can refer to: *Compactly generated group, a topological group which is algebraically generated by one of its compact subsets *Compactly generated space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subspa ..., a topological space whose topology is coherent with the family of all compact subspaces {{Disambiguation Mathematics disambiguation pages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactly Generated Group
In mathematics, a compactly generated (topological) group is a topological group ''G'' which is algebraically generated by one of its compact subsets.. This should not be confused with the unrelated notion (widely used in algebraic topology) of a compactly generated space -- one whose topology is generated (in a suitable sense) by its compact subspaces. Definition A topological group ''G'' is said to be compactly generated if there exists a compact subset ''K'' of ''G'' such that :\langle K\rangle = \bigcup_ (K \cup K^)^n = G. So if ''K'' is symmetric, i.e. ''K'' = ''K'' −1, then :G = \bigcup_ K^n. Locally compact case This property is interesting in the case of locally compact topological groups, since locally compact compactly generated topological groups can be approximated by locally compact, separable metric factor groups of ''G''. More precisely, for a sequence :''U''''n'' of open identity neighborhoods, there exists a normal subgroup In abstract algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactly Generated Space
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subspace ''A'' is closed in ''X'' if and only if ''A'' ∩ ''K'' is closed in ''K'' for all compact subspaces ''K'' ⊆ ''X''. Equivalently, one can replace ''closed'' with ''open'' in this definition. If ''X'' is coherent with any cover of compact subspaces in the above sense then it is, in fact, coherent with all compact subspaces. A Hausdorff-compactly generated space or k-space is a topological space whose topology is coherent with the family of all compact Hausdorff subspaces. Sometimes in the literature a compactly generated space refers to a Hausdorff-compactly generated space. In these cases compactness is often explicitly redefined at the beginning to mean both compact and Hausdorff (and quasi-compact takes the meaning of compact). In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |