Search Results

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 …

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 …

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 …

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, …

Claim: A subspace $S$ of a locally compact Hausdorff space $X$ is closed iff $S \cap C$ is compact for each compact subspace $C$. Proof. The implication $\Rightarrow$ is clear. For the implication $ …

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 …

There are a pair of double covers $\text{SU}(2) \times \text{SU}(2) \to \text{SO}(4) \to \text{SO}(3) \times \text{SO}(3)$, and the first resp. the second more or less reduces the classification of fi …