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Is every topological space homeomorphic to a subspace of a locally contractible space?

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  • $\begingroup$ en.m.wikipedia.org/wiki/Cone_(topology) $\endgroup$
    – Mikael de la Salle
    Commented May 7, 2021 at 5:33
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    $\begingroup$ @MikaeldelaSalle The cone on a Hawaiian earring is contractible (since it is a cone), but not locally contractible or even locally simply connected. $\endgroup$
    – BVQr
    Commented May 7, 2021 at 5:36
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    $\begingroup$ @BVQr: There are different notions of ``locally'' around. Sometimes it means that every point has a neighbourhood with that property, sometimes that there exists a neighbourhood base. $\endgroup$
    – user130903
    Commented May 7, 2021 at 7:23
  • $\begingroup$ @BVQr My bad, I had interpreted your question in the other way. $\endgroup$
    – Mikael de la Salle
    Commented May 7, 2021 at 7:32
  • $\begingroup$ @Zero the definition I use is a space $X$ is locally contractible at a point $x$ if for every neighborhood $U$ of $x$ there is a neighborhood $V$ of $x$ contained in $U$ such that the inclusion of $V$ is nulhomotopic in $U$. A space is locally contractible if it is locally contractible at every point. $\endgroup$
    – BVQr
    Commented May 7, 2021 at 8:34

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