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    Compactness and countable compactness of ordinal spaces $W(\xi)$

    P-WESTOP04-09

    Area
    Topology
    Seen at
    WESLEYAN
    Years
    2004
    Status
    draft
    Contents

    For an ordinal \(\xi\), let \[ W(\xi)=\{\sigma:\sigma<\xi\} \] with its usual order topology. Equivalently, use the subbasis consisting of \[ A(\eta)=\{\sigma:\eta<\sigma<\xi\}, \qquad B(\eta)=\{\sigma:\sigma<\eta\} \] for \(\eta<\xi\).

    1. Determine exactly for which ordinals \(\xi\) the space \(W(\xi)\) is compact.
    2. Determine exactly for which ordinals \(\xi\) it is countably compact.

    Prove both assertions.

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