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Is $D^2$ and the point space $P$ containing a point of $D^2$ homeomorphic? Are the two space of same homotopy type?

I am seeking for a example of two space that are homotopy equivalent but not homeomorphic.

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    $\begingroup$ A point and a line are homotopic (a line is contractible) but not homeomorphic, and they have different cardinalities. $\endgroup$ – Travis Willse Sep 7 '14 at 10:57
  • $\begingroup$ what about $P$ and $D^2$? $\endgroup$ – user151456 Sep 7 '14 at 11:05
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    $\begingroup$ No, they are not homeomorphic. Remember that a homeomorphism is also a bijection... $\endgroup$ – user98602 Sep 7 '14 at 11:10
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Any contractible space is by definition homotopy equivalent to the one-point space, for instance an interval, a disk, the real line, the Euclidean plane, $\mathbb{R}^n$, Bing's house with two rooms, etc. Any space with more than a single point is not homeomorphic to the one-point space because cardinality is a homeomorphism invariant.

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