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    Compactness and separation axioms for four real-line topologies

    P-UNL871F13-PS8.1

    Area
    Topology
    Topics
    Compactness, Separation Axioms
    Status
    draft
    Contents

    For each of the following spaces:

    1. determine whether it is compact; and

    2. determine the largest natural number \(i\) for which it satisfies the separation axiom \(T_i\).

    Consider:

    • \(\mathbb R\) with the excluded-point topology;

    • \(\mathbb R\) with the included-point topology;

    • the lower-limit line \(\mathbb R_\ell\);

    • the subspace \([0,1]\subset\mathbb R_\ell\).

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