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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