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Let us call a topological space $(X,\tau)$ $\mathbb{R}$-like if it is homogeneous, connected, $T_2$, has a basis consisting of open sets homeomorphic to $X$, and $|X|>1$.

What is an example of an $\mathbb{R}$-like space that is not homeomorphic to some power of $\mathbb{R}$?

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    $\begingroup$ $\mathbb{R}^2\setminus\{0\}$? $\endgroup$
    – Qfwfq
    Commented Oct 7, 2019 at 21:01
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    $\begingroup$ Maybe you should modify the question to "not homeomorphic to an open subset of $\mathbb{R}^n$" $\endgroup$
    – Kevin Casto
    Commented Oct 7, 2019 at 21:47
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    $\begingroup$ And all open subsets of all normed spaces, and probably some more TVSs. $\endgroup$
    – YCor
    Commented Oct 7, 2019 at 21:48
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    $\begingroup$ Also $\mathbf{R}^2\smallsetminus\mathbf{Q}^2$ satisfies this and is not locally contractible. $\endgroup$
    – YCor
    Commented Oct 7, 2019 at 21:49
  • $\begingroup$ Thanks for these nice examples! Can the first answerer @Qfwfq put his example in an answer so we can close this thread? $\endgroup$
    – Dominic van der Zypen
    Commented Oct 8, 2019 at 6:46

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An example is $\mathbb{R}^2\setminus\{0\}$

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