Example for a space that is contractible to precisely one of its points Give an example for a space that is contractible to one of its points and is not contractible to another of its points.
I am really curious about that space, I have thought about tree with $n$ vertices but It doesn't seem to work also maybe it is true for comb space or zigzag space but it is just a feeling, so please help me. Thank you very much.
 A: The comb space works although any point on the 'base line' can be deformation retracted to so depending on the interpretation of your question, this might not quite fit your criteria. Instead, we can take a space which is morally the comb space but where we quotient out by the subset of points which can be deformation retracted onto. I believe (though I have not checked) that this space is homeomorphic to the intersection of the unit disk with the union of the set of lines through the origin in the plane with rational slope. Ofcourse, we don't need to take the intersection of this set of lines with the disk, so we could have just taken the set $$\{(x,y)\in\mathbb{R}\mid y=\alpha x, \alpha\in\mathbb{Q}\}\cup\{(x,y)\mid x=0\}.$$
This space only has a single point which can be deformation retracted onto and that is the origin. Any other point has all neighbourhoods containing points on lines which the prescribed point does not lie on, and so any attempt to deformation retract the space to this point must at some point break continuity (this can be made more precise but hopefully the intuition is clear).
