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Just thought of another answer to this. Any topology on a finite set is compact. Any map from a discrete topology is continuous. Hence by the famous theorem on maps from compact spaces into Hausdor …

Presumably you don't want to allow arbitrary non-expansive maps, otherwise you could simply take that. One thing that you could do artificially is to take the subcategory of "metric spaces + nonexpan …

I'd like to recast Reid's (excellent) answer slightly. The essence of it is the following principle: To show that a limit or colimit doesn't exist in some category, embed your category in one whe …

Picking up on Gerald's interpretation of the question (namely, that it really focusses on infinite dimensional vector spaces) then I say: absolutely not! For example, piecewise-smooth paths in some E …

This isn't a complete answer, but I think that whatever the family is, it contains compact metric (metrisable) spaces. With a paracompactness argument, I suspect that it would extend to locally compa …

I work with infinite dimensional manifolds so am extremely distrustful of anything that requires some sort of compactness condition. Most of the time, it's just too restrictive. Consider a really ni …

(I was going to leave this as a comment but decided that it's a bit long for that) A couple of remarks: You express an aversion to Riemannian metrics because you want to be able to apply this in th …

I first learnt about the Atiyah-Singer index theorem from Shanahan's Springer notes (638). I liked it because while developing the main theory, it went through the standard examples (Dirac, Dolbeaut, …

In a locally convex topological vector space, say $E$, there are a few other notions of convergence besides the standard one which are sometimes of use. Mackey-convergence: there is a bounded, absolu …

This is the swindle, isn't it? There's an elegant way to phrase this with lots of sines and cosines, but working it all out is too much like hard work. Here's the quick and dirty way. Let $T: S^\in …

Here's a picture of a smash product that I drew for this talk, as far as I can tell it's what "Qfwfq" is describing in the middle paragraph.

My instinct (need to sit down with a piece of paper to confirm it) is No and No. For the first, I'm pretty sure that any compactly generated non-compact Hausdorff space will provide a counter-example …

For connected locally Euclidean spaces, paracompactness is equivalent to second countable. Therefore if we truly insisted that our manifolds be second countable then the only thing that we would lose …

Just a minor addition to the other answers, since you seem particularly interested in cohomology theories and operations on them. We often concentrate on operations of a particular cohomology theory, …

Use the source, Luke. Specifically, chapters 14 (Smooth Bump Functions) to 16 (Smooth Partitions of Unity and Smooth Normality). You may be particularly interested in: Theorem 16.10 If $X$ is Li …