TITLE: What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?
QUESTION [61 upvotes]: Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) force us to work very hard to prove the main theorems setting up all of the big machinery to get the payoff we want (e.g. invariance of domain, fixed point theorems). But why should I care about these arbitrary and terrible spaces and functions in the first place when, as far as I can tell, any manifold which occurs in applications is at least piecewise-differentiable and any morphism which occurs in applications is at least homotopic to a piecewise-differentiable one? 
In other words, do topological manifolds really naturally occur in the rest of mathematics (without some extra structure)?

REPLY [8 votes]: I learned a lot from the answers and discussion above. Here is my personal view, not necessarily applicable to others.
I published my Lectures on Algebraic Topology (W. A. Benjamin) in 1967. The book is  an extensive elaboration of notes from a course I taught. All the manifold theorems in my book refer to topological manifolds - I only make a few side remarks about differentiable manifolds, De Rham cohomology and PL-manifolds. My book was later expanded and slightly revised by John Harper, renamed as Algebraic Topology: A First Course (Perseus, 1981).
I was delighted, in the process of teaching that course, to learn that so much could be proved in this subject without bringing in analysis and with very little messy (imho) combinatorial argumentation. Purity of method is an essential part of mathematical aesthetics, as Hilbert emphasized. And to achieve that purity, one usually has to discover new techniques that apply in greater generality than the previously known methods.
I was glad to see that J. P. May, in his lovely 1999 treatise A Concise Course in Algebraic Topology (U. of Chicago Press), also focused on topological manifolds.