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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user151456
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3A point and a line are homotopic (a line is contractible) but not homeomorphic, and they have different cardinalities. – Travis Willse Sep 07 '14 at 10:57
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what about $P$ and $D^2$? – user151456 Sep 07 '14 at 11:05
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1No, they are not homeomorphic. Remember that a homeomorphism is also a bijection... – Sep 07 '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.
Dan Rust
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