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This is not directly an answer to any of your questions as stated but a riff on the theme of "what does $\overline{\mathbb{F}_p}$ look like?" The best answer to this question I've found so far comes f …

I hope I'm using the terminology correctly. What I mean is this: fix $K = \mathbb{R}$ or $\mathbb{C}$ (I'm interested in both cases). Which topological spaces $X$ have the property that for every o …

There is an example at PlanetMath of a Hausdorff space which is not completely Hausdorff / functionally Hausdorff. On the other hand it is second-countable, hence first-countable and hence compactly …

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective o …

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 …

In general, we might ask when we can find interesting spaces $X, Y$ such that $X$ is homeomorphic to $[X, Y]$. By the Lawvere fixed point theorem $Y$ must have the fixed point property. Happily, $Y = …

Assuming that $G$ is discrete, the homotopy quotient $X/G$ fits into a fiber sequence $$X \to X/G \to BG$$ and hence, by the long exact sequence in homotopy, its fundamental group $\widetilde{G} = \ …

When a certain kind of homotopy theorist says "space," they don't mean a topological space, or even an object which in any sense has an underlying topological space. The simplest translation of what " …

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 …

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 …

One way to get $G/H$ from the map $f : H \to G$ is to first deloop it, getting $Bf : BH \to BG$, and then take homotopy fibers, getting a fiber sequence $$H \to G \to G/H \to BH \to BG.$$ This sugge …

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 …

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 …

All of the Galois connections I know involving a power set arise from a relation $R : X \times Y \to 2$ as described for example here. As you observe, this relation can often be used to define a topol …

Some comments, unrelated to the other answer, on what happens in the case of the Zariski topology. Here the extra axiom is $$V(I(S_1 \cup S_2)) = V(I(S_1)) \cup V(I(S_2))$$ (where, to fix notation, …