20
votes

### Why are extremally disconnected spaces so hard to give examples of?

We can always get non-principal ultrafilters on sets from non-discrete extremally disconnected spaces.
Let $X$ be a topological space. Let $\text{Clo}(X)$ denote the Boolean algebra of clopen subsets ...

12
votes

Accepted

### Topos notions coming from topology and uniqueness of generalizations

If the absence of adjoints is what worries you, you can consider this to be a two-step process - and I would argue that in practice this is the case in the vast majority of cases:
One first generalize ...

9
votes

### On a metrized $n$-dimensional manifold $X$, does every $x \in X$ have a small ball $B_\delta(x)$ that is homeomorphic to $\mathbb R^n$?

Consider $f:\mathbb R\to\mathbb R$, $f(x)=x\sin(1/x)$ with $f(0)=0$. Then $$d(x,y)=|x-y|+|f(x)-f(y)|$$
defines a metric on $\mathbb R$ which induces the usual topology (because of the continuity of $f$...

6
votes

### How to properly define a slice knot (or a locally flat disk)?

The question is, what do you want to achieve by local flatness?
One possible answer is the following: You want your slice disk to have a tubular neighborhood. To make sure that you have a tubular ...

6
votes

Accepted

### On connected sum of compact manifolds along a submanifold

Note that "closed" means "compact without boundary".
Here are answers.
(1): Yes, if $M_1$ and $M_2$
are closed manifolds then $M_1\sharp_KM_2$ is a closed manifold. If
$M_1$ and $...

6
votes

Accepted

### Perfectly normal but not collectionwise normal space in ZFC

Bing's Example H is found in
R. Bing, Metrization of Topological Spaces, Canadian J. Math. 3 (1951), 175-186.
It is on the page following his more famous Example G. It is constructed in ZFC and has ...

6
votes

Accepted

### Can a scattered profinite set continuously surject onto a non-scattered profinite set?

No this is not possible. There might be an easier proof than this but I don't know one off the top of my head.
First note that a profinite set is scattered if and only if no non-empty closed subspace ...

5
votes

### Category of topological spaces with open or closed maps

A nice recent paper of Bezhanishvili and Kornell (which actually references this MathOverflow question specifically) has shown that the existence of products fails very badly in $\mathrm{Top}_{\mathrm{...

5
votes

### How algebraic can the dual of a topological category be?

I have not properly digested your framework of definitions; the following example may or may not fit within it. (If not, it would be helpful to explain why not.)
Let $\operatorname{CH}$ be the ...

4
votes

### Menger and Scheepers subsets of $\mathbb R$

I believe that the sets in the paper that you cite (subsets of the Cantor space that contain the set of eventually 0 elements) are homeomorphic to sets of reals that contain the rational numbers, by a ...

4
votes

### A finite dimensional continuum with a subset $A$ such that both $A$ and $X\setminus A$ are dense and contractible

A fancy decomposition of the closed unit disk $X$ of $ \mathbb{C}$ into two dense contractible sets. Let $D_1$, $D_2$ be a $2$-partition of the interval $[0,1]$ into dense sets. Consider the ...

4
votes

### A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$

Umm, $A$ is path-connected and it is fairly simple -- just change the coordinates one by one. Say we want to go from $u\in A$ to $v\in A$. On $[0, \frac{1}{2}]$ change linearly $u_1$ to $v_1$, then on ...

4
votes

Accepted

### Stone-Čech compactification of a Boolean subalgebra of $\{0,1\}^S$

A partial answer.
for your specific example, the algebra $B$ generated by the rational intervals, the answer is negative. This algebra is countable and dense in $2^\mathbb{R}$. It is countable and ...

4
votes

Accepted

### Is any submetrizable linear topology linearly submetrizable?

Yes. Let $(X,\tau)$ be a toplogical vector space and $d$ a metric on $X$ generatin a coarser topology $\pi$. Recursively, one finds balanced $\tau$-neighbourhoods $U_n$ of the origin with
$$ U_{n+1}+...

4
votes

### Can a scattered profinite set continuously surject onto a non-scattered profinite set?

Here's the Stone duality perspective:
Scattered profinite sets (a.k.a. scattered Stone spaces) correspond under Stone duality to superatomic Boolean algebras: $B$ is superatomic if every quotient ...

3
votes

### How algebraic can the dual of a topological category be?

Check out the following papers.
Barr and Pedicchio, $\text{Top}^\circ$ is a quasi-variety
Barr and Pedicchio. Topological spaces and quasi-varieties.
Dimov, Pedicchio, and Tironi. Frames and Grids.
...

3
votes

### On a metrized $n$-dimensional manifold $X$, does every $x \in X$ have a small ball $B_\delta(x)$ that is homeomorphic to $\mathbb R^n$?

Regarding the first question: if the metric is the Carnot-Caratheodory metric on step-two sub-Riemannian manifolds, small balls should be homeomorphic to standard balls.
(It is known that Carnot-...

3
votes

Accepted

### Continuous collections of arcs

Let $F(c)=f^{-1}(c)$, $F: C\to K(X)$, where $K(X)$ is the hyperspace of $X$.
Then $F$ is continuous by your condition and thus $F(C)$ is compact. Consequently $\bigcup F(C)$ is compact as well and $\...

2
votes

### Universally closed implies proper for locales

It turns out that Vermeulen has essentially answered the question in [A note on stably closed maps of locales].
The argument there implies:
Theorem.
Let $g : X \to S$ be a morphism of locales.
The ...

2
votes

### A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$

For the first question, yes, $S \setminus A$ is contractible. Rewrite the standard basis so that it is indexed by $\mathbb{Z}$. Let $U$ be the bilateral shift. Then $U$ is an isomorphism on $\ell^p$ ...

2
votes

### Stone-Čech boundary is not extremally disconnected

The question has already been answered satisfactorily, but I think the following presentation, which is implicit in the second part of YCor's answer, will clarify things. The fact that $\beta\mathbb{...

2
votes

### Does regular $G_\delta$ imply normal?

Edit: I made a mistake in disagreeing with Henno, who after all is right. But perhaps someone can learn from my mistake.
It is worth noting that a space in which every closed set is "regular $G_\...

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