nLab hemicompact space


A topological space XX is called hemicompact iff there exists a sequence (K n) n(K_n)_{n\in\mathbb{N}} of compact subsets K nXK_n\subseteq X such that X= nK nX=\bigcup_{n\in\mathbb{N}} K_n and K nint(K n+1)K_n \subseteq int(K_{n+1}) holds.


If XX is hemicompact and (K n)(K_n) a sequence as in the definition, then every compact subset KXK\subseteq X is contained in one of the K nK_n. In other words: {K nn}\lbrace K_n \mid n\in\mathbb{N}\rbrace is cofinal in the set of all compact subsets (partially ordered by inclusion)

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