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We know that when we prove that two topological spaces are homeomorphic to each other in fact we are proving that these spaces are in fact equal under deformations.

Why? this question is very intriguing for me.

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No we're not! What do you mean by deformation anyway? –  Clive Newstead Nov 6 '12 at 18:52
    
@CliveNewstead for example, a torus can't be deformed in a sphere because they aren't homeomorphic to each other. –  user42912 Nov 6 '12 at 18:54
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It's not because they're not homeomorphic to each other; there are homeomorphic spaces between which there is no deformation. I think you might be thinking of something like a homotopy equivalence, rather than a homeomorphism. –  Clive Newstead Nov 6 '12 at 18:55
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That gives you one direction, but you can’t get the other direction. It’s not even clear what it would mean to be a deformation of the irrationals into $\omega^\omega$, for instance, yet those two spaces are homeomorphic. –  Brian M. Scott Nov 6 '12 at 18:56
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That much is true, provided that you define deformation carefully enough. For example, as @Clive mentioned, you need to rule out deformations that squeeze a line segment to a point, even though they can be carried out in a nice, continuous fashion. –  Brian M. Scott Nov 6 '12 at 19:04
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