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(Marc Kegel posted his answer just before I posted this. I will leave it because perhaps this helps elaborate some of the points) No, this is not true in general. Here is an example of what can ha …

No, there is no generalization to "n-arcwise connected" that you ask for. Take $X= \mathbb{R}^3$. This space is as nice a space as you could ever hope for. It is also contractible, so in particular …

Let M be a (non-compact) smooth manifold and consider the set $C^\infty_c(M)$ of smooth real-valued functions with compact support. We can give this function space several topologies. Here are four: …

If a CW-complex is contractible, then it strongly deformation retracts onto the inclusion of a point. However for general spaces it is well-known that just because a space is contractible, it does n …

I have seen it claimed that (for compactly generated Hausdorff spaces) the geometric realization of the singular (internal) simplicial space is homotopy equivalent to the original space. I know how to …

The topology of topological monoids can be arbitrarily bad. Here is an instructive example. Let X be any topological space. Then we will construct a (commutative) topological monoid M whose underlying …

As Zhen Lin points out in the comments to your question, the category of compact Hausdorff spaces has all small colimits. However the inclusion functor from CHTop to Top does not preserve these colimi …

Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying …

The category of topological spaces has a forgetful functor to set which commutes with both small limits and colimits (it has both a left and a right adjoint). Moreover Set is a Grothendieck topos and …

Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will …

Let X be a (finite dimensional) manifold. Consider smooth mapping space $$PX = C^\infty(I, X)$$ where I = [0,1] is the closed interval. Is this space paracompact? What if we fix a point x in X and con …

There are a couple different versions of the geometric realization of simplicial spaces. There is the literal one, which is badly behaved in general. Then there are better realizations, e.g. the "fat" …

My favorite example of a space which is not homotopy equivalent to a CW complex is the Long Line. All it's homotopy groups vanish (exercise 1) but the long line is not contractible (exercise 2). It's …

There is a Wikipedia entry on topological modular forms, where you can see further references. The primitive version of Topological modular forms is as a generalized cohomology theory. Like K-theory a …