Search the corpus

    A quotient of a compact space is compact

    E-ENMFU

    Area
    Topology
    Topics
    Compactness, Quotient Spaces
    Status
    draft

    Show that a quotient of a compact space is again compact.

    Exercise

    Solution

    1. Let \(q:X\to X/{\sim}\) be the quotient map. It is continuous and surjective.

    Continuity is part of the definition of the quotient topology.

    2. If \(X\) is compact, then \(X/{\sim}=q(X)\) is compact.

    The continuous image of a compact space is compact.

    Source Collections