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    A set open in a subspace but not in the ambient space

    E-NZ4WI

    Area
    Topology
    Topics
    Point-Set Topology, Subspace Topology
    Status
    draft

    Give an example of spaces \(A\subseteq B \subseteq X\) such that \(A\) is open in \(B\) but \(A\) is not open in \(X\).

    Solution

    No: Take \([0, 1] \subset [0, 1] \subset \RR\). Then \([0, 1]\) is tautologically open in \([0, 1]\) as it is the entire space, But \([0, 1]\) is not open in \(\RR\):

    • E.g. \(\theset{1}\) is not an interior point (every neighborhood intersects the complement \(\RR\setminus[0, 1]\)).

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