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Is it possible to find a continuous bijection $f: \mathbb{R} \rightarrow \textrm{Z}$ where $\textrm{Z}$ is a compact Hausdorff space? ($\mathbb{R}$ endowed with usual topology)

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up vote 4 down vote accepted

Yes. For example, you can take $Z$ to be a wedge of two circles, and let $f$ wrap first around one circle, then the other.

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Nice counterexample. If I understand your construction correctly, the same topology could be obtained be taking the usual topology on $\mathbb R$ and change local basis at 0 to be the usual neighborhoods plus complement of a bounded closed interval. (Or, equivalently, take the one-point compactfication of reals and make a quotient space by identifying $\infty$ and 0. – Martin Sleziak Aug 9 '11 at 11:11

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