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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 …

I'd like to clear up something that came up in the comments. There are two natural ways to fit the finite cyclic groups together in a diagram. One is to take the morphisms $\mathbb{Z}/n\mathbb{Z} \t …

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 …

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} = \ …

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 $ …

Every totally ordered set naturally gives rise to a topology; the basis of the topology is the set of open intervals and open rays, just as in the order definition of the topology on R. See the Wikip …

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 " …

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 …

Some Google-fu turned up an example here.

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 …

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 …

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 …

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 …

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 …

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 …