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Can we have a continuous bijection from $[0,1]$ to $[0,0.5)\cup (0.5,1]$?

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    $\begingroup$ Can you find some topological property that one of the spaces has but the other hasn't? $\endgroup$ – Daniel Fischer Aug 21 '14 at 21:16
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    $\begingroup$ Can you make some connection between the image of the set and the preimage? $\endgroup$ – IAmNoOne Aug 21 '14 at 21:17
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    $\begingroup$ IVT $\endgroup$ – WimC Aug 21 '14 at 21:20
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No, because a continuous function sends connected sets into connected sets (connection is a topological property).

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It is not possible because the set $[0,0.5) \cup (0.5, 1]$ is the union of two disjoint closed sets. So this space is not connected.

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  • $\begingroup$ You mean 'open' subsets of $[0,1]$, not closed sets, don't you? $\endgroup$ – Berci Aug 21 '14 at 22:21
  • $\begingroup$ @Berci, I was referring to it as a subspace of $\mathbb{R}$. $\endgroup$ – IAmNoOne Aug 27 '14 at 21:16

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