# Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

4,177
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### Is there a literature name for this concept of a "graded metric"?

Given a space $X$, I have been thinking about a function $d\colon X \times X \times \mathbb{N} \to \mathbb{R}_{\geq 0}$ (i.e. with values that are nonnegative reals) with the properties below. One may ...

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### A metric characterization of Hilbert spaces

In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...

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### Is there some kind of construction of a "canonical unirational variety" like the one for toric varieties?

Toric varieties in some sense a "canonical rational variety" in that one can construct them from purely combinatorial data and this combinatorial data makes it possible to turn many ...

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### Existence of diffeomorphism interpolating affine map and identity

$\newcommand{\R}{\mathbb{R}}$Suppose $\Omega$ is a bounded, convex domain in $\R^{m}$. Fix $x_1, x_2\in\Omega$ and an invertible matrix $A\in\mathrm{GL}^{+}(m)$ with positive determinant.
Let $U\...

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### When can we find a net, defined on a totally ordered index set, converging to a non-isolated point in a compact Hausdorff space?

Let $X$ be a compact Hausdorff space and $p\in X$ be a non-isolated point. Is it always possible to find a net $(x_\alpha)_{\alpha\in (I,\leq)}$ in $X\setminus\{p\}$ converging to $p$ such that $(I,\...

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### The graph topologies for powersets

Given a topological space $X$ and a metric space $(Y,d_Y)$, there are a number of topologies one may put on the space $\mathcal{C}(X,Y)$ of continuous functions from $X$ to $Y$. Perhaps the most ...

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### A ZFC example of a star-$K$-Menger space which is not star-$K$-Hurewicz

An open cover $\mathcal U$ of a space $X$ is said to be $\gamma$-cover if $\mathcal U$ is infinite and for each $x\in X$, the set $\{U\in\mathcal U : x\notin U\}$ is finite.
A space $X$ is said to be ...

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### Trivial convergent sequences in $\beta X$

Let $X$ be a Tychonoff space and denote by $\beta X$ its Stone-Čech compactification. We know, for example, that if $X$ is an $F$-space then $\beta X$ is an $F$-space and, therefore, in $\beta X$, the ...

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### References (and a question) on the "fine" topology of powersets

Recently I've been trying to understand powerset topologies better, and came upon the following reference:
Frank Wattenberg, Topologies on the set of closed subsets. Pacific J. Math. 68(2): 537-551 (...

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### A ZFC example of a Menger space which is not Scheepers

$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...

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### Closed subset of unit ball with peculiar connected components

Let $n\geq 2$ and denote by $B\subset \mathbb{R}^n$ the closed unit ball.
Does there exist a closed subset $A\subset B$ containing $0\in \mathbb{R}^n$ with the following properties i,ii,iii?
i) $\{0\}$...

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### Describing a time-varying process with a manifold

I am a beginner in topology and I am trying to define a model for some computations. My questions are speculative:
I am wondering what is the proper way to add time in a manifold so as to describe a ...

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### Changing a metric to that 2 points have different distance

Let $X$ be a compact metric space. Assume that $X$ has more than $2$ points (or even better, that $X$ is connected with more than 1 point). Given a metric $d$ on $X$ we define $$d(x,X)=\max\{d(x,z):z\...

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### A metric characterization of the real line

Is the following metric characterization of the real line true (and known)?
A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real ...

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### Neighborhoods of idempotents in topological inverse semigroups

In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse ...

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### When did derivative algebras first appear?

In the paper "The Algebra of Topology" (Annals of Mathematics, 45, 1944), McKinsey and Tarski proposed derivative algebras (p183) to define the derive set in topology as follows.
Suppose $K$ ...

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### The pro-discrete space of quasicomponents of a topological space

Let $X$ be a topological space.
Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$.
It is not hard to check that $P^X : \textbf{...

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### Do compactly generated spaces have a more direct definition?

Is there an elementary way to define Haussdorf-compactly generated weakly Hausdorff topological spaces in a way that does not need defining topological space first?
Weakly Hausdorff sequential spaces ...

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### Non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides

I need to construct an example of two non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides. Spaces should have induced ("good&...

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### Do germs of open sets around a point form a frame?

Let $X$ be a topological space and $x \in X$ a point. Let $\Omega$ be the set of open sets (viꝫ. the topology) of $X$, and $\Omega_x$ the set of germs around $x$ of open sets, that is, $\Omega_x = \...

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### Closed graph correspondence which never contains the whole support

Let $I=[a,b]$ with $a<b\in\mathbb{R}$ and denote by $\mathcal{M}(I)$ the set of Borel probability measures on $I$ equipped with the topology induced by the weak convergence of measures.
Does there ...

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### Infinite tensor/Fubini product of ultrafilters

Given an infinite family $\{\mathcal{F}_{\lambda}$, $\lambda <\kappa\}$, $\kappa \geq \omega_0$, of (ultra)filters of a set $X$, how it is defined the infinite tensor/Fubini product $$\bigotimes_{\...

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### Constructively valid reference for the soberness of discrete spaces and points of a locale coproduct

I am looking for constructively valid references for the following two related facts:
discrete topological spaces are sober,
the points of a locale coproduct are the disjoint union of the points of ...

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### Naturality of the map from $\pi_1(X)\to \pi_1(K(G,1))$

The following result is from Hatcher Proposition 1B9.
Let $X$ be a connected CW complex and let $Y$ be a $K(G,1)$. Then every homomorphism $\pi_1(X,x_0)\to \pi_1(Y,y_0)$ is induced by a map $(X,x_0)\...

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### Maximally fine topologies on $B(H)$ making the unit ball compact

Let $H$ be a Hilbert space, and $B(H)$ its algebra of bounded operators. One of the reasons the Ultraweak topology is (in a way) more useful than the weak operator topology is that the Ultraweak ...

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### Does the category of locally compact Hausdorff spaces with proper maps have products?

nlab presents a proof that the category of locally compact Hausdorff spaces does not admit infinite products in general. In particular it shows that there is no infinite product of $\mathbb{R}$, since ...

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### Existence of a nice-ish topology on the powerset of a topological space

This is a follow-up question to my previous question, Existence of a *really* nice topology on the powerset of a topological space, which, in a few words, asked about whether given a topological space ...

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### Homological restrictions on certain $4$-manifolds

I am not very familiar with the non-compact $4$-manifold theory. So I apologize if the following question is very silly.
Let $X$ be a non-compact, orientable $4$ manifold that is homotopic to an ...

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### Can the set of compact metrisable topologies naturally be equipped with the structure of a standard Borel space?

Let $X$ be a compact metric space, and let $K_X$ be the set of non-empty closed subsets of $X$, equipped with the $\sigma$-algebra
$$ \mathcal{B}(K_X) \ := \ \sigma(\{C \in K_X : C \cap U = \emptyset\}...

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### When is this topology compatible with the partial ordering?

This question was first asked here, on math stack exchange, but wasn't able to attract any attention. Now that I am thinking more, it feels like the most suitable place for this question is here.
...

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### A question about uniformities generated by pseudometrics

Suppose that for all $n$ natural numbers, $d_{n}$ is a pseudometric on set $X
$. Define $d=\sum_{n=1}^{\infty }a_{n}\frac{d_{n}}{1+d_{n}}$, where $\left(
a_{n}\right) $ is a sequence of positive ...

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### Compact Hausdorff spaces as a cocompletion of profinite sets

It is well-known that the category CH of compact Hausdorff spaces has a strong categorical flavor (e.g. Properties of the category of compact Hausdorff spaces, which includes Manes' theorem asserting ...

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### Can continuous correspondence be represented via continuous functions?

Let $\Theta \subset \mathbb{R}^n, \mathcal{X} \subset \mathbb{R^m}$, and suppose that $C: \Theta \rightrightarrows \mathcal{X}$ is a correspondence defined by $f: \Theta \times \mathcal{X}\to \mathbb{...

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### A a question about the metrization of uniform spaces

I have read two theorems about the metrization of uniform spaces from Engelking and Kelley.
Kelley's condition (b) is slightly different from Engelking's corresponding result for Vi's.
I think these ...

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### Classification of closures of additive subgroups of $\mathbb{R}^n$

If $G$ is an additive subgroup of the real numbers $\mathbb{R}$ and $\overline{G}$ is the topological closure of $G$ then either
$\overline{G} = a \cdot \mathbb{Z}$ for some $a \in \mathbb{R}$, or
$\...

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### Existence of disintegrations for improper priors on locally-compact groups

In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...

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### Weight of the compactification of a topological space

I am wondering the following question. If $w(X)$ denotes the weight of a Tychonoff space $X$, that is the least infinite cardinal $\kappa$ for which $X$ has a basis of cardinality $\kappa$, and $Z$ is ...

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### Powersets of simplicial sets vs. powersets of topological spaces

Motivation. Recently I've been trying to understand how well- or ill-behaved are the many different powerset topologies one can put on the powerset of a topological space, and in particular I'm trying ...

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### A simple homotopy lemma [migrated]

I'm trying to prove a lemma, suppose that $F:[0,1]×[0,1]→\bar{\mathbb{D}}$, a continuous function from the square to the closed disc, such that $F(t,0)=e^{2iπt}$, and in the other three borders, F is ...

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### Hereditarily Lindelöf spaces with density continuum

Since there are L-spaces (provably in ZFC), under CH we have regular, hereditarily Lindelöf spaces with density continuum. However, I cannot find an example of such a space under not CH, nor a proof ...

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### Perfectly normal compactification of weak-star dual of Banach space

Let $X$ be an infinite-dimensional (otherwise the answer to my question below is trivial) separable real Banach space with topological dual $X^*$, and denote by $\sigma(X^*,X)$ the weak-star topology ...

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### Existence of a *really* nice topology on the powerset of a topological space

TL;DR. Given a topological space $X$, is there a natural way to "induce" a topology on $\mathcal{P}(X)$ from the topology of $X$ in such a way that 1) all the basic operations of set theory (...

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### Niceness properties of quotient spaces by continuous equivalence relations

Given an equivalence relation $R$ on a topological space $X$, there are certain conditions we may ask of $R$ that imply certain well-behavedness conditions on the quotient space $X/\mathord{\sim}_R$. ...

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### How to construct large, hereditarily separable compact spaces?

In [1] Fedorcuk, using diamond, proved that there is a hereditarily separable compact space of cardinality $2^{2^\omega}$.To my best knowledge, Kunen created a humanly digestible proof, but he has ...

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### Making the powerset into a topological monoid

Every monoid $X$ induces a monoid structure $\circledast$ on $\mathcal{P}(X)$ via
$$U\circledast V := \{uv\ |\ u\in U,v\in V\}.$$
Moreover, a morphism of monoids $f\colon X\to Y$ induces a morphism of ...

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### Boundaries of subsets of simply-connected domains

I am trying to prove the following assertion: Let $B$ be a simply-connected set, and let $B' \subsetneq B$ be a proper open connected subset. Then, there exists a point $x \in \partial B'$ of the ...

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### Analogue of open/closed maps for measurable spaces

$\newcommand{\A}{\mathcal{A}}\newcommand{\T}{\mathcal{T}}$The notions of continuous map of topological spaces and measurable function of measurable spaces are very similar:
A map of topological ...

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### Does "Invariance of domain" true for injective Darboux function (instead of continuous injection)?

$f:U\subset \mathbb{R^n}\to\mathbb{R}^n$ be an injective Darboux map.
Does this implies $f$ is an open map?
If $f$ is continuous then the result follows from "Invariance of domain ".
My ...

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### (When) can you embed a closed map with finite discrete fibers into a (branched) cover?

Assume all spaces are topological manifolds. A branched cover is a continuous open map with discrete fibers. A finite branched cover is one with finite fibers.
Questions. Given closed map $X\to S$ ...

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### Characterization of the Scheepers property by Scheepers game

$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...