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    The equalizer of two maps into a Hausdorff space is closed

    E-DWC4M

    Area
    Topology
    Topics
    Continuous Functions, Separation Axioms
    Seen at
    WESLEYAN
    Years
    2014
    Status
    draft

    Let \(f, g: X \to Y\) be continuous; assume that \(Y\) is Hausdorff. Show that \(\ts{x \mid f(x) = g(x)}\) is closed in \(X\).

    Solution

    Let \[ E=\{x\in X:f(x)=g(x)\}. \] The map \[ F:X\to Y\times Y,\qquad F(x)=(f(x),g(x)) \] is continuous. Since \(Y\) is Hausdorff, its diagonal \[ \Delta_Y=\{(y,y):y\in Y\} \] is closed in \(Y\times Y\). Hence \[ E=F^{-1}(\Delta_Y) \] is closed in \(X\).

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