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The simplest case - where you only need the Banach fixed point theorem - is quite beautiful if you think about it the right way: your map lands somewhere on the land it marks, so somewhere on the map …

Here is a situation where you really use this coalgebra structure (which, as other answers have mentioned, exists over a field in particular). If $X$ is a homotopy associative $H$-space, then $H_{\b …

The triangle inequality is natural. In any setting where the metric is related to some kind of optimization problem, for example if $d(a, b)$ measures the "length" of the "shortest path" between point …

This is an attempt to synthesize ideas that have appeared in other answers, for example sigfpe's and Tim Perutz's. Feel free to edit if you think the ideas can be better expressed. The idea I want t …

Here's a boring reason, and it may or may not convince you: any function $f : X \to Y$ between topological spaces has the property that the preimage of the entire space $Y$ is the entire space $X$, an …

The answer is that this never happens for manifolds which are of finite type in the sense that they are homotopy equivalent to finite CW complexes. Serre showed that a simply connected finite CW compl …

Take straight lines connecting the points $(t, t^2, t^3), t \in \mathbb{N}$. As far as I can tell you can also boost this to $t \in \mathbb{R}$. The point here is that two distinct lines between poi …

This is an elaboration on Todd Trimble's comment about Tom Leinster's lovely posts about codensity monads. I quite like the codensity monad story; here is my preferred way of telling it. Suppose you …

The two-point discrete space already doesn't have a (universal) connectification, in the sense that two points don't have a coproduct in the category of connected spaces. If $X$ were such a coproduct, …

Open sets can be identified with maps from a space to the Sierpinski space, and maps out of a space pull back under morphisms. (In other words, if you believe that the essence of what it means to be …

This is form 8L of the axiom of choice at http://consequences.emich.edu/CONSEQ.HTM, and is known to be equivalent to countable choice. The proof is fairly straightforward: if $B_1, B_2, ...$ is a cou …

It would be best to talk about the category of sets first, I think. Any isomorphism class of sets shows up so many times that a given isomorphism class doesn't itself form a set - for example, $\{ 1, …

Asking when a continuous map $f : X \to Y$ has a continuous section is analogous to asking when a Diophantine equation over $\mathbb{Z}$ has a solution over $\mathbb{Z}$; see, for example, this blog p …

I don't think it's a good idea to mix the slogans "topological spaces are locales" and "topological spaces are $\infty$-groupoids." I think the former slogan encapsulates what we ended up defining as …

Terence Tao's cheap nonstandard analysis can be interpreted as taking place in a topos related to the topos $\text{Set}^{\mathbb{N}}$ of sets indexed by the natural numbers; see this math.SE question …