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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 topological space X is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different a ..., 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 alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactly Generated Space
In topology, a topological space X is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other. Also some authors include some separation axiom (like Hausdorff space or weak Hausdorff space) in the definition of one or both terms, and others do not. In the simplest definition, a ''compactly generated space'' is a space that is coherent with the family of its compact subspaces, meaning that for every set A \subseteq X, A is open in X if and only if A \cap K is open in K for every compact subspace K \subseteq X. Other definitions use a family of continuous maps from compact spaces to X and declare X to be compactly generated if its topology coincides with the final topology with respect to this family of maps. And other variations of the definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |