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    Locally compact Hausdorff spaces are regular

    E-L4NKO

    Area
    Topology
    Topics
    Compactness, Separation Axioms
    Status
    draft

    Show that every locally compact Hausdorff space is regular.

    Solution

    Let \(X\) be locally compact Hausdorff, let \(A\subset X\) be closed, and let \(x\notin A\). The open set \(U=X\setminus A\) is a neighborhood of \(x\). By the compact-closure neighborhood lemma from §29, there is a neighborhood \(V\) of \(x\) such that \[ \overline V\text{ is compact and }\overline V\subset U. \] Then \[ V\quad\text{and}\quad X\setminus\overline V \] are disjoint open sets containing \(x\) and \(A\), respectively. Thus \(X\) is regular.

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