27
votes

### Why study finite topological spaces?

Long comment:
Note that a finite topological space is nothing but a pre-order (order, iff the topology is $T_0$), given by $x\leq y$ iff $x\in \overline{\{y\}}$. An equivalence relation on a ...

23
votes

### Why study finite topological spaces?

As Uri Bader's answer notes, finite topological spaces are equivalent to partially ordered sets (posets). Now, combinatorialists who are interested in the topology of finite posets most commonly study ...

14
votes

### Why study finite topological spaces?

Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa:
Are finite spaces a model for finite CW-complexes?
So, when it comes to computing homotopy ...

10
votes

### Why study finite topological spaces?

And what would be some instances where 'standard circumlocutions' used to avoid them?
For example, one can define contractability and compactness (for metrizable spaces) in terms of maps of finite ...

9
votes

Accepted

### Which maps of topological spaces have the right lifting property with respect to all split monomorphisms?

SEE THE EDITS BELOW
The answer is basically: fibrations equivalent to trivial bundles. (Dylan Wilson speculated about this in the [comments (https://mathoverflow.net/questions/453780/which-maps-of-...

5
votes

Accepted

### Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds

I would like to argue that the situation considered in the comments is "close to generic".
Let $(M,g)$ be a Lorentzian manifold that is not strongly causal; this implies that $(M,g)$ is also ...

4
votes

Accepted

### Is the projective limit $\mathcal{D}(\mathbb{R})$ separable?

The projective topology topology you describe is coarser than the usual inductive limit topology on $\mathscr D(\mathbb R)$ (the universal properties imply that it is enough to have continuity of the ...

4
votes

### Pointwise convergence and disjoint sequences in $C(K)$

I claim that either of your properties (sequences or nets) is equivalent to having only finitely many non-isolated points.
Property $S$: Any pointwise null sequence in $C(K)$ has an almost disjoint ...

4
votes

Accepted

### Is a local diffeomorphism with nice boundary values a diffeomorphism?

This is true and follows from a more general fact. Note that in the dimension $n=2$ the unbounded component of $f(\partial\mathbb{D})$ is simply connected.
Theorem. Let $f: \bar{\mathbb{B}}^n \to \...

4
votes

Accepted

### Variation of concept of a Lusin space

The closed graph theorem implies that nice comparable topologies on a vector space coincide. More precisely, If $(X,\sigma)$ is a barrelled locally convex space (e.g., a Fréchet space) and $\tau$ is a ...

3
votes

Accepted

### Cohomology of the amplitude space of unlabeled quantum networks

$H^*(Y_n;\mathbb{Q}) \cong (H^*((S^3)^N;\mathbb{Q}))^{\Sigma_n}$.
Hence with rational coefficients this reduces to analyzing the invariant subspace of the action of $\Sigma_n$ on $H^*((S^3)^N;\mathbb{...

2
votes

Accepted

### Embedding of half open half closed $n$-set in $n$-space

The answer is yes. It suffices to show that $h$ restricted to each of the sets $1\leq |x|\leq 3/2$ and $3/2\leq |x|<2$ is an embedding, because the image of points near the boundary $|x|=2$ will be ...

2
votes

### Topological characterisations of properties of posets

I will discuss
Question 1: Is there a nice purely topological characterisation when a connected finite topological space with $T_0$ corresponds to a lattice and when this lattice is distributive?
...

2
votes

### Stone-topological/profinite equivalence for quandles

Here is how the proof works for monoids. This doesn't really answer the question so I made this communitywiki, but maybe the OP will find it useful.
Let $X$ be a compact Hausdorff space. Call an ...

Community wiki

2
votes

Accepted

### A topological characterisation of a.e. continuity

I adapt the proof of a more general result: Theorem 4.12 in Chapter XI of K. Kuratowski and A. Mostowski’s book “Set Theory: with an introduction to descriptive set theory” (see page 408).
Claim. $f$ ...

1
vote

### Topological characterisations of properties of posets

Recall that an Alexandrov space is a topological space where the intersection of arbitrary collection of open sets is open. Alexandrov duality states that the category of Alexandrov spaces is ...

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