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

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0
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2answers
53 views

A question about open subsets of Hilbert space whose complements are compact sets

Let $H$ be an infinite-dimensional separable Hilbert space. Let $C$ be the intersection of a denumerably infinite sequence of sets, each of which is the complement of a compact subset of $H$. In other ...
0
votes
0answers
42 views

Pro-constructible subset of scheme intersects very dense subsets?

Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$? If $Y$ is just constructible, this is true. ...
3
votes
1answer
757 views

Who coined “mob” and “clan” and why these words?

A mob is a word used for a topological semigroup which is a Hausdorff space. A clan is a compact connected mob with a two-sided identity element. Who used these words with these meanings first and ...
9
votes
1answer
417 views

Can you write $\mathbb R^2$ as a disjoint union of two totally disconnected sets?

Can you write $\mathbb R^2$ as a disjoint union of two totally disconnected sets?
3
votes
1answer
185 views

Equivalence relation defined by the existence of a homeomorphism

Let $(X,\tau)$ be a topological space. We assign to $(X,\tau)$ an equivalence relation $\simeq_{(X,\tau)}$ in the following way: $x\simeq_{(X,\tau)} y$ if and only if there is a homeomorphism ...
1
vote
1answer
60 views

A countable tight topological group where every countable subset is metrizable

I am looking for an example of a topological group with countable tightness with the property then it is not metrizable, but every countable subset is metrizable but I cannot construct an example. ...
29
votes
1answer
720 views

Are there only countably many compact topological manifolds?

Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be ...
-4
votes
0answers
45 views

Example on Non Hausdorff Space [closed]

Consider R with the following topology τ ={(a,∞):a∈R}∪{∅,R}. Is the topology Hausdorff?
1
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0answers
43 views

What is the quotient $\mathbb{T}^3/\mathbb{Z}_2$? [migrated]

What is the quotient space $\mathbb{T}^3/\mathbb{Z}_2$, When $\mathbb{Z}_2$ acting on 3-dimensional torus $\mathbb{T}^3 := \mathbb{R}^3/\mathbb{Z}^3$ by sending $x$ to $-x$? Does anyone know how to ...
0
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0answers
50 views

Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space

Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map ...
4
votes
1answer
231 views

Can an abelian group have a minimal group topology?

In the abstract of this paper, it is said that a minimal group topology on an abelian group is not Hausdorff. Suppose $G$ is an abelian group and $\mathcal T$ is a minimal group topology on $G$ and ...
1
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0answers
73 views

Set nor its compliment contain an uncountable closed set [closed]

Does there exist a set $X$ subset of the real numbers such that no uncountable closed set is contained in $X$ or $X^c$?
-1
votes
0answers
42 views

Universal covering space of CW complex has CW complex structure [migrated]

We know that covering spaces of many low dimensional CW complexes such as graph (CW structures with only 0-cells and 1-cells) and all compact surfaces has CW structure. [The second fact is due to ...
1
vote
0answers
102 views

Closed subsets of Souslin orders [migrated]

A total order without end-points is Souslin if it is complete, non-separable and ccc. Such orders may or may not exist but when they do, there can be a vast zoo of them. Can we get consistent ...
1
vote
2answers
106 views

Is the poset of all precompact group topologies on an abelian group $G$, order-isomorphic to $\operatorname{Sub}(\hat{G})$?

In this page, in abstract, it is claimed that the poset of all Hausdorff precompact group topologies on an abelian group $G$, is order-isomorphic to the the subgroup lattice of $\hat{G}$, the ...
1
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0answers
36 views

Local section of Lie Groupoids

Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...
5
votes
1answer
184 views

$\text{Cont}(X,X)$ and $\neg\mathsf{GCH}$

For a topological space $(X,\tau)$ let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$. Is it consistent that there a space $(X,\tau)$ such that $$|X| < |\text{Cont}(X,X)| < ...
2
votes
3answers
294 views

Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$

Let $(X,\tau)$ be a topological space. Let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$. What can be said about spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$? For ...
-3
votes
1answer
107 views

partial converse of existence of covering spaces [closed]

Suppose $X_1$ and $X_2$ are two spaces which has a common finite sheeted covering space $Z$ (may be distinct index)...from here can we conclude that there exists a space $W$ such that $X_1$ and $ X_2$ ...
6
votes
4answers
473 views

Inserting an open and simply-connected set between a compact set and an open set

In a paper I am reading, the following is considered obvious: Let $K$ be a compact and connected subset of $\,\mathbb R^2$, with $\mathbb R^2\smallsetminus K$ also connected, and $U\subset \mathbb ...
2
votes
3answers
459 views

Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the ...
6
votes
1answer
430 views

Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?

It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In ...
12
votes
2answers
314 views

Counterexample for associativity of smash product

In Section 1.7 of Parametrized Homotopy Theory by May and Sigurdsson it is stated that the smash product of pointed topological spaces is not associative (which is just another hint that ...
5
votes
2answers
147 views

Is the strong Whitney topology connected?

$\def\bbR{\mathbb R}\def\ssp{\kern.4mm}$Put more precisely, let $C$ be the set of all continuous functions $f:\bbR\to\bbR$ when $\bbR$ has its standard order topology. Let $\mathscr T$ be the set of ...
0
votes
0answers
44 views

Is the core of an atom in lattice of group topologies a coatom?

Let $(G,\mathcal T)$ be an abelian topological group such that for any nontrivial group topology $\mathcal S$ on $G$ with $\mathcal S\subseteq \mathcal T$ we have $\mathcal S = \mathcal T$. Let ...
0
votes
0answers
30 views

Definition of submaximal (group) topology

By [3] (or [2]), a submaximal group topology on a group is the infimum of all maximal group topologies on it. By [4], a submaximal topological space is one with all dense subsets open. By [1], a ...
1
vote
1answer
104 views

A Hausdorff atom in lattice of group topologies

Do you have an example of an infinite Hausdorff nonabelian topological group $(G,\mathcal T)$ such that for any nontrivial group topology $\mathcal S$ on $G$ with $\mathcal S\subseteq \mathcal T$ we ...
1
vote
1answer
147 views

Two questions on path connected spaces

Is it true to say that a compact hausdorff space $X$ is path connected if and only if for every continuous function $f:X\to \mathbb{C}$, we have $f(X)\subset \mathbb{C}$ is path connected? ...
0
votes
1answer
141 views

Totally non hereditary $C^{*}$-subalgebras

Assume that $B$ is a $C^{*}$ subalgebra of $A$. We say $B$ is totally non hereditary subalgebra of $A$ if not only $B$ is not a hereditary subalgebra but also it is not isomorphic to any ...
5
votes
2answers
203 views

Quotient of metric spaces

Let $(X,d)$ be a compact metric space and $\sim$ an equivalence relation on $X$ such that the quotient space $X/\sim$ is Hausdorff. It is well known that in this case the quotient is metrizable. My ...
2
votes
1answer
98 views

Is $ {C_{c}}(G) $ a meager subset of $ {L^{2}}(G) $ for a second-countable locally compact Hausdorff group $ G $?

The following problem is a stumbling block in a research project that I am working on: Problem. Let $ G $ be a second-countable locally compact Hausdorff group with a fixed Haar measure. Is it ...
1
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0answers
39 views

Why finite dimensional MCS-space implies $\mathbb{R}^m\times cone$ locally?

Frequently, I see a statment of the result in reference that a finite dimensional Alexandrov spaces is locally a product $\mathbb{R}^m\times cone$. It seems that this result comes from Perelman's ...
4
votes
1answer
145 views

Proper continuous image of metrizable space

Motivated by the following post, "Gelfand duality" and the fact that "a Hausdorff continuous image of a compact metric space is metrizable", we ask: What is a counter example of two locally ...
4
votes
1answer
89 views

Some questions about “inspecting” the boundary of a closed ball in Hilbert space

Let H be a separable Hilbert space and suppose that H is infinite dimensional. Let B be a closed ball of H-which has a positive radius-and let S be the boundary of B. A non-empty subset C of H is an ...
4
votes
2answers
508 views

Co-Hausdorffification

Given a topological space $(X,\tau)$ we can define the "$T_2$-ification" of $X$ by setting $T_2(X,\tau) = X/\simeq$ where $x\simeq y$ in $X$ if and only if for every open neighborhood of $x$ has ...
6
votes
1answer
220 views

A question on compact sets

Let $K\subset \mathbb{R}^N$ be a compact set. We say $K$ is "good" if the following property holds: Given a set of open neighborhoods $\{x\in U_x\subset \mathbb{R}^N\}_{x\in K}$ there exists a finite ...
1
vote
0answers
110 views

Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
1
vote
0answers
100 views

Properties of “incomplete finite simplicial complexes”

Definition: We say that $K'$ is an incomplete finite simplicial complex if there exists a finite simplicial complex $K$ such that $|K'|=|K|\backslash Y$ where $Y$ is a union of some open faces of K. ...
10
votes
1answer
192 views

Choosing a metric in which homeomorphism is Holder continuous

Let $X$ be a compact metrizable space, and let $f:X \to X$ be a homeomorphism. Is it always possible to choose a compatible metric on $X$ in which $f$ is Holder continuous? I've tried some simple ...
5
votes
2answers
108 views

Image of poset with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology $\tau_{\text{int}}(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = ...
3
votes
1answer
62 views

Product of posets with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\leq x\}$ and ...
1
vote
0answers
84 views

continuous homomorphism with open image in a product topological space

I originally posted the problem below in MathSE, but I deleted it since it is not receiving any attention. So I decided to transfer the question here instead. Let $G$ be a profinite group and $I$ ...
6
votes
0answers
95 views

Is $(\omega+1)^\omega$ with the box topology ultraparacompact?

Let $\omega+1$ be endowed with the interval topology, that is $U\subseteq (\omega+1)$ is open if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite. We call $U\subseteq (\omega+1)$ basic if ...
3
votes
1answer
148 views

Paracompact zero-dimensional space without clopen partition refinement

If $(X,\tau)$ is a topological space we say that an open cover $\mathcal{U}$ is a clopen partition cover if it consists of disjoint clopen sets. Trivially, every clopen partition cover is locally ...
2
votes
0answers
80 views

Regularity of Dirac measure on Baire sets [closed]

Suppose $X$ is a locally compact Hausdorff space. Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$, to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$. ...
6
votes
2answers
261 views

The Metrizability of Symmetric Products of Metric Spaces

The (infinite) symmetric product of a based topological space $(X,e)$, denoted by $\mathrm{SP}(X,e)$, can be viewed as the topological space of ''multisets'' in $X$ containing the base point $e$ ...
6
votes
1answer
316 views

Sierpinski-like spaces

Let $\mathbb{S}$ be the Sierpinski space, that is $\mathbb{S}$ has $\{0,1\}$ as a base set, and $\tau = \{\emptyset, \{0\}, \{0,1\}\}$ as a topology. The Sierpinski space $\mathbb{S}$ has the ...
2
votes
1answer
108 views

Closed sets in ordinal spaces

I'm studying the ordinal space $[0,\kappa[$ where $\kappa\neq \omega$ is a cardinal of countable cofinality and I want to know why there are in $[0,\kappa[$ two disjoint closed sets of cardinality ...
-2
votes
1answer
305 views

What is the meaning of non-Hausdorff spaces in algebraic geometry [closed]

At the beginning I think that I should warn everybody reading this post: I don't know much about algebraic geometry so most of specialists in this subject may seen my question as ignorant. To be ...
2
votes
1answer
236 views

A question in general topology

Motivated by this question we ask: Up to homeomorphism, are there only a finite number of connected locally compact hausdorff topological space $X$ such that $X$ has an open set $U$ ...