Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
3
votes
1answer
135 views
Question about of comeager set
If $G\subseteq2^{\mathbb{N}}$ is comeager then exist is a partition $\mathbb{N}=A_0\cup A_1$, $A_0\cap A_1=\emptyset$, and sets $B_i \subseteq A_i$ for $i \in \{0,1\}$, such that for $A \subseteq ...
1
vote
0answers
117 views
Sum-epimorphisms and prod-monomorphisms
Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
...
0
votes
1answer
72 views
Totally non fixed point property
Edit: According to the comment of Pietro Majer, I revise the question
Is there a non singleton compact connected Hausdorff topological space $X$ for which the following property hold?:
"Constant ...
8
votes
1answer
208 views
Translates of meager sets
Does there exist a meager set of reals M such that every meager set can be covered by countably many translates of M? This is the category analogue of the following.
10
votes
1answer
234 views
When is $X \rightarrow \text{Spec}(C(X))$ a homeomorphism?
Let $X$ be compact Hausdorff topological space. Consider the ring $C(X)$ of continuous functions $X \rightarrow \mathbb C$ (we do not consider the C* algebra structure, just consider $C(X)$ as a ring) ...
0
votes
1answer
127 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
63 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
798 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 ...
20
votes
4answers
986 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
189 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
62 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.
...
31
votes
1answer
763 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 ...
1
vote
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
votes
0answers
52 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
232 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
vote
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
112 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
vote
0answers
37 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
190 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
4answers
354 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
110 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
477 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
470 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
433 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
320 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
150 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
31 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
148 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
147 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
207 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
99 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
vote
0answers
40 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
90 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
221 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
112 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
85 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
96 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
150 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
262 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 $e$ ...
