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Hot answers tagged gn.general-topology

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 ...
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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 ...
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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 $f$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}+...
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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 ...
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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. ...
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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-...
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