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It is well-known that every connected, locally connected compact metrizable space $X$ contains an arc, that is, a subspace homeomorphic to $[0,1]$. Are there topological properties we can add to these hypotheses to ensure that $X$ contains a subspace homeomorphic to $[0,1]^2$?

My naive guess is to demand that $X$ has covering dimension $\dim(X)\geq 2$ but I wouldn't be surprised if this is still insufficient.

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    $\begingroup$ Yes, this is insufficient, Pontryagin surface would be an example. $\endgroup$
    – Moishe Kohan
    Commented Apr 24, 2023 at 18:02
  • $\begingroup$ @MoisheKohan: I gather a Pontryagin surface is a certain connected sum of infinitely many projective planes, and relevant properties are that it is 2-dimensional, homogeneous and not locally contractible (hence cannot contain a 2-disk). What is a good reference for the properties? $\endgroup$
    – Igor Belegradek
    Commented Apr 24, 2023 at 18:43
  • $\begingroup$ Iff there exists a quotient map from $R^{2}$ onto a subspace of $X$ such that each point preimage is a non separating continuum, and so that the point preimages form an upper-semi-continuous decomposition of the plane. There is some redundancy in this answer if $X$ has a reasonable topology. $\endgroup$
    – Paul Fabel
    Commented Apr 24, 2023 at 20:20
  • $\begingroup$ If $X$ is metric then $X$ contains a disk iff there exists a map from $R^{2}$ onto a subspace $Y$ of $X$, such that compacta in $Y$ have compact preimage, and so that the preimage of each $y$ in $Y$ does not separate the plane. $\endgroup$
    – Paul Fabel
    Commented Apr 24, 2023 at 20:36
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    $\begingroup$ @IgorBelegradek: I think "A useful functor and three famous examples in topology" by Williams or "Degree-one, monotone self-maps of the Pontryagin surface are near-homeomorphisms" by Daverman and Thickstun. $\endgroup$
    – Moishe Kohan
    Commented Apr 24, 2023 at 21:47

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