TITLE: a example of two space that are homotopy equivalent but not homeomorphic
QUESTION [7 upvotes]: 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.

REPLY [8 votes]: 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.