# Questions tagged [gn.general-topology]

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

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### Continuity of the Restriction Map Between Function Spaces

Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as
\begin{align}
\rho:&C(X,Y)\rightarrow C(Z,...

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53 views

### When does a function space with compact-open topology have countable chain condition?

As in title，when a function space with compact-open topology has countable chain condition？ Are there some sufficient and necessary conditions？ Who give some references about this topic？
McCoy and ...

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**1**answer

48 views

### Dimension of a topological space equals the supreme of the dimension of its open cover

For a topological space $X$ which is covered by a family of open subsets $\{U_i\}$, then show that $\dim(X)=\sup (\dim(U_i))$.
I understand that $\dim(X)\geq \sup(\dim(U_i))$, so it only suffices to ...

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79 views

### Quotient space homeomorphic if the complements are homeomorphic? [on hold]

Let $A_{0}$ be a closed subspace of $A$ and $B_{0}$ a closed subspace of $B$ such that there exists a
$i:A\rightarrow B$ a continuous injective map (or even a continuous embedding) such that the ...

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106 views

### What does the Grothendieck topos tell us about the homotopy type of a space?

Let $M_1$, $M_2$ be two closed connected topological manifolds. We can consider the small sites of open coverings of them, and the categories of sheaves on these sites.
what can we say about $M_1$ ...

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28 views

### Problem on Topological Spaces [on hold]

I have to solve this problem, in particular the point of the union of an arbitrary family of elements that belongs to the topology.
This is the Text: "Let {p} be an arbitrary singleton set such that ...

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45 views

### Connectedness of sequence spaces (countable products) in different metrics

My question concerns a quite elementary problem in set-theoretic topology: Assume that $(X,d)$ is a compact metric space. Consider the infinite product $X^{\mathbb{Z}}$ of all (two-sided) sequences in ...

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**1**answer

106 views

### Comparison of several topologies for probability measures

Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...

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**1**answer

176 views

### Is there a universally meager air space?

Let $\mathcal P$ be a family of nonempty subsets of a topological space $X$. A subset $D\subset X$ is called $\mathcal P$-dense if for any $P\in\mathcal P$ the intersection $P\cap D$ is not empty.
A ...

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55 views

### Making a quasi-compact open into an affine open

Let $X$ be a spectral topological space, $U\subset X$ be a quasi-compact open subspace. Is there necessarily some scheme structure on $X$ (we do not require it to be affine) such that $U$ endowed with ...

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59 views

### On product of Borel sigma algebras [closed]

First of all I'm not a native English speaker so please bear with me. I have to prove that $B(\mathbb{R}^l)\otimes\ B(\mathbb{R}^d)=B(\mathbb{R}^{l+d})$ is true for $l,d \in \mathbb{N}$
So I know ...

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**1**answer

250 views

### A Besicovitch-type Covering Theorem

In the book The Geometry of Domains in Spaces by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M_\mu f$, where $\mu$ is a Radon measure, using their ...

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**1**answer

366 views

### Quotient of Three Dimensional Torus by Permutation on Coordinates

The Mobius Strip can be realized as a quotient of $T = (S^1)^2$ via the identifications $(x,y) \sim (y,x)$.
I tried to generalized this concept to a higher dimension, and consider the quotient of $(...

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**1**answer

37 views

### Hausdorff quotient collapsing and separating a prescribed collection of disjoint closed subsets

Let $X$ be a compact Hausdorff space (I don't mind assuming it's metrizable).
Let $A_i$ $i\in \mathbb{N}$ be a collection of disjoint closed subsets of $X$.
My question: Does there exist a Hausdorff ...

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88 views

### The underlying space of open dense subscheme

Let $X$ be an affine scheme such that there exists a field $k$ and a morphism of finite type $X\rightarrow \mathrm{Spec}\,k$. Let $U\subset X$ be an open dense subscheme. Is the underlying space of $U$...

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264 views

### The underlying space of an affine open dense subscheme

Let $X$ be a Noetherian scheme, $U\subset X$ be an affine open dense subscheme. Is the underlying space of $U$ necessarily homeomorphic to the underlying space of $X$?

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139 views

### Is unitary group paracompact?

In this paper Martin Schottenloher notices that the unitary group $U(H)$ of a separable Hilbert space $H$ is metrizable in the strong operator topology. As a corollary (see R.Engelking, 5.1.3), it is ...

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32 views

### Every second-countable, uncountable Hausdorff space contains a non-empty countable subspace with no isolated points [migrated]

Could you give me a hint on how to solve the following problem?
Every second-countable, uncountable Hausdorff space
contains a non-empty countable subspace with no isolated points.

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78 views

### Reference request: lower sets of a preorder form a lattice

Consider a set $S$ with a preorder $\preceq$ (a preorder is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $...

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177 views

### Which points in the Samuel compactification of a metric space $X$ are limits of uniformly discrete subsets of $X$?

Given a metric space $(X.d)$ the Samuel compactification of $X$, written $sX$, is the unique compactification with the property that if $Y$ is an arbitrary compact Hausdorff space and $f:X\rightarrow ...

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90 views

### Is there a homogeneous compactum where non-empty $G_\delta$s have non-empty interior?

A space $X$ is called an almost $P$-space if $Int(G) \neq \emptyset$ for every non-empty $G_\delta$ subset $G \subset X$.
Every $P$-space (that is, a space where $G_\delta$s are open) is an almost $P$...

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61 views

### Closed and discrete sets

Let $\kappa$ in an uncountable regular cardinal and $X$ be a space and $e(x)=\kappa$, where the ``extent'' $e(X)$ of $X$ is the supremum of the cardinalities of
closed discrete subsets of $X$. My ...

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97 views

### Quasi-compactification of locally spectral spaces

Let $X$ be a locally spectral topological space (i.e. a space admitting an open cover by spectral spaces). Does there necessarily exist a quasi-compact locally spectral space $Y$ and an injective ...

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98 views

### Countable union of Menger spaces

A topological space $X$ has Menger's property $\textsf{S}_{\mbox{fin}}(\mathcal{O}, \mathcal{O})$ if, for each sequence of open covers, $\mathcal{U}_1, \mathcal{U}_2, \cdots $, we can select finite ...

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238 views

### Again, proving that specific preorder on the set of measurable functions is symmetric

This question is followup to the previous similar question. There I was trying to find good sufficient condition for abstract preorder to be symmetric, but now, as I have found good formalization of ...

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58 views

### Is there a Lindelof $P$-space which is not discretely generated?

A space $X$ is:
Lindelof if every open cover for $X$ has a countable subcover.
A $P$-space if every $G_\delta$ subset of $X$ is open.
Discretely generated if for every non-closed set $A \subset X$ ...

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52 views

### Is there an $L$-space whose square is selectively $d$-separable?

An $L$-space is a hereditarily Lindelof regular space which is not separable.
A space is $d$-separable if it contains a dense set which is the countable union of discrete sets.
An $L$-space can't ...

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69 views

### source of argument about relative primeness of simple closed curves on tori

I have know this argument for decades. I have no idea of its source. If anyone knows (not guesses) its origin, then I would be very appreciative. My guesses are among Ralph Fox, JHC Whitehead, RH ...

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122 views

### Topological spaces with (possibly non-uniquely) unique scheme structure

This question consists of 5 related sub-questions.
Does there exist a non-empty scheme $X$ such that any reduced scheme whose underlying space is homeomorphic to that of $X$ (possibly via a ...

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107 views

### A scheme whose underlying space is the product of the underlying spaces of schemes

We know that the product of two spectral topological spaces is spectral.
If $X$ is the underlying space of the scheme $\mathrm{Spec}\,\mathbb{Z}[x]$, what is a simple example of an affine scheme ...

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604 views

### Which definition of “proper” is better?

It is well known that topology and algebraic geometry assign different meanings to the word "proper".
Let us recall the relevant definitions from topology (and we work in the context of topological ...

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246 views

### The (fiber of the) cofiber of the fiber of a map of spaces

Consider a fiber sequence of spaces
$$F \overset{i}{\to} E \to B$$
The cofiber $C(i)$ of the inclusion of the fiber comes with a canonical map $C(i) \to B$. Its possible to show (using some point ...

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157 views

### What's the minimal weight of a maximal space?

A non-empty topological space without isolated points is called maximal if every finer topology on that space has at least an isolated point. The existence of a (Hausdorff) maximal space is a simple ...

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222 views

### Grothendieck topology on a scheme equivalent to the circle

Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a ...

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97 views

### Homeomorphic extension to totally disconnected sets

Dear Mathoverflow Community,
I am looking for a reference for the following topological fact:
Fact
Let $E$ and $F$ be two totally disconnected compact subsets of the plane (can assume perfect if ...

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81 views

### Is there a linearly Lindelof space which is not weakly Lindelof?

Recall that a space is:
"Lindelof", if every open cover has a countable subcover.
"Linearly Lindelof", if every open cover which is linearly ordered by $\subseteq$ has a countable subcover.
"weakly ...

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92 views

### Borel selections of usco maps on metrizable compacta

The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...

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168 views

### A non-Borel union of unit half-open squares

On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$
Observe that for every $z\in \mathbb C$ and $p\in\{0,1,2,3\}$ the set $...

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156 views

### What is the smallest number of hyperplanes covering $\ell_2$?

For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.
By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...

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40 views

### Decomposition with given closures [closed]

Let $(X,\mathcal T)$ be a topological space. About the subsets $A,B,C$ of $X$ it is known that $$\mathrm {cl} (C)= A \cup B\,, \quad \mathrm {cl} (A) = A\,, \quad \mathrm {cl} (B) = B\,.$$
Does it ...

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239 views

### On projectively countable sets in the Hilbert cube

A subset $A$ of a topological space $X$ is called projectively countable if for any continuous map $f:X\to\mathbb R$ the image $f(A)$ is countable.
It is easy to see that each projectively countable ...

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245 views

### A slightly canonical way to associate a scheme to a Noetherian spectral space

Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...

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### Topological properties of Noetherian affine schemes that do not hold for general Noetherian spectral spaces

I used to think that the only reason why an affine scheme with a Noetherian space can fail to be Noetherian is nilpotents. It turns out that this is not true.
This leads me to the following question:...

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107 views

### Point-free topology, but with $\sigma$-algebras instead of spaces

I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had:
If abstract $\sigma$-algebras (i.e. certain boolean ...

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222 views

### Fractal plane continuum with order $\aleph_0$?

Continuum means compact and connected.
Define the order of a point $x$ in a continuum $X$ to be the least cardinal $\alpha$ such that $X$ has a neighborhood base of open sets at $x$ with no more ...

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116 views

### Countably compact non-compact perfect spaces

Recall that a space is countably compact if every infinite set has an accumulation point. A space is perfect if every closed set is a countable intersection of open sets. One of the classical ...

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46 views

### How to determine the family of bounded functions from an infinite Fort space to $[0,1]$?

Definition: Let $X$ be a topological space and $b\in X$. We call $X$ a Fort space (with particular point $b$), when $X$ has topology $\{A\subseteq X: b \not\in A \; \text{or} \; X\setminus A\; \text{...

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53 views

### Quantification over Nets

On a topological space $X$, a net is defined as a map $A \ni \alpha \longmapsto x_{\alpha} \in X$ from a directed set $A$.
With this, compactness of $X$ (for instance) is equivalent to "every net $(...

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260 views

### Do continuous maps factor through continuous surjections via Borel maps?

Let $f \colon X \twoheadrightarrow Y$ be a continuous surjection between compact Hausdorff spaces, and $g \colon \mathbb{R} \to Y$ a continuous function. Can you always find a Borel-measurable ...

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56 views

### Version of Knaster-Tarski that gives unique fixed point

Can someone give a reference (if it exists) to a version of Knaster-Tarski that gives a unique fixed point?