pqnelson
commented
9 years ago A brief answer (perhaps cryptic and probably unsatisfactory), "deformation" usually refers to Homotopy. Who cares?
Well, we can say two topological spaces are "the same" (homeomorphic) if there is a continuous bijection between them. That's hard to do in practice.
Nuts! What can we do to simplify this process of checking if two spaces are "the same" or not?
Well, if two spaces are "the same", then they have the same topological properties. If $X$ and $Y$ are homeomorphic topological spaces, and $X$ is connected, then necessarily $Y$ is connected. But if $Z$ is a disconnected topological space, then there is no hope of being homeomorphic to $X$.
This approach (of assigning topological properties to topological spaces) is done in algebraic topology. We can classify the "different types" of homotopic paths using a group (the homotopy group!).
So I understand, at a beginner's level, Topology from two different perspectives: Collections of subsets satisfying the axioms, and continuous deformations. So the joke goes, to a topologist most shapes are spheres. But how is that connected to the subset notion of a topology?
I have a few guesses but they're all rough. One is that the definition of continuity used in topology, that of a function which inversely maps open sets to open sets, allows us to define a continuous deformation. We often use this on Euclidean spaces to obtain the sorts of continuous deformations that make intuitive sense.
But also in the case of a finite topology, sets being "open" allow us to draw the points and then draw "circles" or "enclosures" around the sets of points that are open. These circles we can also continuously deform in the intuitive sense so long as we don't allow them to cross other points ... I'm not sure how much sense this makes.