Q:

Is it correct to think about homeomorphisms as deformations?

The definition of homeomorphism is that of a continuous bijection with continuous inverse. Because we can think of continuous functions as functions that maps nearby points to nearby points, we could think that a homeomorphism is a function that in some sense deforms one topological space into another.
But I'm a little unsure of that intuition because I know there's something called homotopy. I haven't get to homotopy yet, but by what I've heard, it seems that the true mathematical concept of "deforming one space into another" has something to do with homotopy rather than homeomophisms.
Is this intuition of homeomorphisms correct? If not, what should be the intuition? Just a way to identify topologies? Thanks very much in advance!

A:

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. "No tearing" implies continuity, and "no gluing" implies bijectivity. While thinking intuitively, it is probably best to restrict oneself to compact Hausdorff spaces, where a bijective continuous function is automatically a homeomorphism (ie. the inverse is automatically continuous).
A homotopy, on the other hand, is also a deformation, but need not respect the "no gluing" condition. For instance, the closed unit interval $[0,1]$ is homotopic to a point! In constructing this homotopy, one does not "tear" the interval, but one does "contract it continuously".
Not sure if that made any sense :)

