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    The diagonal map $\Delta(x)=(x,x)$ is continuous

    E-BOLN3

    Area
    Topology
    Topics
    Continuity, Product Topology
    Status
    draft

    Show that the diagonal map \(\Delta(x) = (x, x)\) is continuous.

    Solution

    1. The diagonal is the product map \((\operatorname{id}_X,\operatorname{id}_X):X\to X\times X\).

    Both component maps are continuous. By the universal property of the product topology, a map into \(X\times X\) is continuous exactly when both coordinate maps are continuous.

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