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My question is stated in the title. As a brief background, I'd like to say I know next to nothing about Topology. The little bit I was exposed to came as an aside in my Multivariate Calculus class; we were discussing which boundary-less surfaces may we apply Stokes' Theorem to and conclude that the $\int_\Sigma (\nabla\times \vec{F})\cdot d\vec{S}=0.$ We mentioned that any surface that is "topologically equivalent" to a sphere, torus, "torus with two holes", etc., fits the bill (i.e., we classified by "genus", or number of holes). We further said that each of these surfaces (with different number of holes) is "topologically different" because one would need to "rip" or "tear" the surface to get another, but in this "game" we only allow "stretching" or "twisting" as the means of deformation.

I've also heard, in passing, the terms "smooth" and "differentiable" applied in the context of Topology. My question is how that comes into Topology. If, as according to my very limited understanding, Topology is the study of "deforming" figures according to certain rules, how does differentiability enter the picture? Furthermore, aren't we considering all figures, i.e., even figures that go beyond the notion of a "function" and would not necessarily be able to be differentiated? Perhaps since my knowledge is so vague on the topic, I'm missing an obvious fact. Anyways, thanks in advance.

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This might be worth reading. The existence of exotic spheres demonstrated that differential topology is not simply another method for topology. –  Michael Greinecker Apr 25 '13 at 19:50
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Yes, there is no real sense of differentiability in general topology, only continuity. Most often, differentiability is talked about for smooth manifolds. –  Thomas Andrews Apr 25 '13 at 19:52
    
The idea "Topology is the study of "deforming" figures according to certain rules" is really just an introductory idea to give an intuition of how topology relates to rigid geometries. Topology is a much broader topic, and that start g idea is just the beginning, not the end. –  Thomas Andrews Apr 25 '13 at 19:57
    
@MichaelGreinecker : thanks for that read! as it turns out, my professor who taught me what I mentioned in my post had John Milnor as his advisor on his dissertation! –  Coffee_Table Apr 25 '13 at 20:16
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What does differentiability do for you anywhere? Amongst other things, it lets you do stuff like talk about velocity - physics makes sense on a differentiable manifold, whereas it is not really possible to do physics in a plain manifold. If you see discussions of string theory, with large dimensions, they are talking about differentiable manifolds, I believe. –  Thomas Andrews Apr 25 '13 at 20:57
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I might as well answer this.

Differentiability has nothing to do with general topology. The definition you are using:

Topology is the study of "deforming" figures according to certain rules

is a useful motivation for topology, but it is just a starting idea. A truer definition for topology is "studying continuity."

Differentiability makes sense only if we can talk about rates of change, and general topology doesn't even have a notion of "direction."

In general topology, we can define the concept of a manifold. An $n$-manifold is a space that looks "locally" like Euclidean $n$-space. For example, a sphere is a $2$-manifold, because if you are at a point on the sphere, the other points near can be given coordinates. Given pairs of real numbers, you can map them (continuously) to a unique points near you, and that map reaches all points near you. We call these local coordinates a "chart."

This still doesn't get us differentiation on the sphere. If you and I are on two points near each other, you and I also might have wildly different charts. In particular, for the points that are "near" both of us, what looks like a "smooth" curve or function to you in your coordinates might look "not smooth" in my coordinates. For "smooth" to make sense on the whole manifold, these charts have to agree about what is smooth.

So we have to add a structure to a manifold to make it differentiable. Essentially, if $S$ is the set of points near both you and me, then the map from my coordinates for the points in $S$ to your coordinates for the points in $S$ has to be differentiable for us to be able to talk about "smoothness" on the manifold.

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I would say this is an extremely deep question, and I admit I cannot give a good/ short answer. But to begin with one might want to look at Topology from the Differentiable Viewpoint, which does not assume much from the reader, and continue with Differential Forms in Algebraic Topology, which might need a little background in algebra and topology.

According to my limited knowledge, nothing answers your question better than Milnor and Bott & Tu.

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