Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,432
questions
2
votes
0
answers
67
views
a connected geometrically unibranch algebraic stack of finite type over a field is irreducible
Let $f:X\to \mathfrak{X}$ be a smooth presentation of geometrically unibranch connected algebraic stack by a scheme, which is geometrically unibranch since being geom. unibranch is local in smooth ...
3
votes
1
answer
136
views
On the Menger property and the Alexandroff duplicate
Recall that a space $X$ is Menger if for each sequence $(\mathcal{U}_n)_{n\in\omega}$ of open covers of $X$, there is a sequence $(\mathcal{V}_n)_{n\in\omega}$ such that, for each $n\in \omega$, $\...
16
votes
1
answer
461
views
Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?
Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \...
1
vote
1
answer
80
views
Is the class of rc-spaces closed under products?
Let $(X,\tau)$ be a topological space. A retraction is a continuous map $r:X\to X$ such that $r$ is the identity on $\text{im}(r)$. We call $S\subseteq X$ a retract of $X$ if there is a retraction $r:...
0
votes
0
answers
129
views
Cyclic group action and finite invariant set
Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$
Is it true that the ...
3
votes
0
answers
105
views
"Practical" references on mapping spaces as infinite-dimensional manifolds
I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
6
votes
0
answers
212
views
Every Polish space is the image of the Baire space by a continuous and closed map, reference
The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501)
Every Polish space (i.e. every separable ...
1
vote
1
answer
99
views
Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?
Let $(E, d)$ be a Polish space and $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P_p (E)$ the space of all Borel probability ...
1
vote
1
answer
90
views
Conditions that ensure the metric topology of $E$ coincides with the initial topology induced by a collection of real-valued functions on $E$
Let $(E, d)$ be a metric space and $\mathcal F$ a collection of real-valued functions on $E$. Assume that for all $x,x_n \in E$ with $n\in \mathbb N$,
$$
x_n \to x \iff [f(x_n) \to f(x) \quad \forall ...
9
votes
1
answer
202
views
Is there a connected Hausdorff anticompact space that is countably infinite?
Cross-posted from MSE.
Following Bankston - The total negation of a topological property, a topological space is called anticompact if all its compact subsets are finite. The linked MSE post above ...
2
votes
1
answer
110
views
A question about a realcompact space and upper semicontinuous function
Nancy Dykes says in the proof of Theorem 3.4 in her article Generalizations of realcompact spaces that by a result of
John Mack, if for every $p\in \beta X\setminus X$ there exists a nonnegative
upper ...
2
votes
0
answers
158
views
Concrete description of “DeMorganian” open sets
Let me begin with a few definitions. My question will be basically how to simplify them to something more manageable. The motivation for these definitions is given at the end.
Let $X$ be a ...
5
votes
0
answers
227
views
Does Tychonov's theorem directly imply Zorn's lemma?
This question was formerly posted on MSE https://math.stackexchange.com/questions/4578923/ without getting an answer.
I know that Tychonov's theorem, Zorn's lemma, the axiom of choice, the well-...
6
votes
1
answer
123
views
For which $X$ is $X\times I$ collectionwise normal?
Many normality-type properties can be characterised in terms of products with the unit interval $I=[0,1]$. For instance, if $X$ is a Hausdorff space, then;
$X$ is normal and countably paracompact if ...
6
votes
0
answers
96
views
Classification of contractible open n-manifolds which embed in a compact n-manifold
Does there exist a classification of contractible open $n$-manifolds ($n\geq 3$) which embed in a compact $n$-manifold? More general, does there exist a classification of contractible open $n$-...
2
votes
1
answer
70
views
Why are the selection principle $S_\text{fin}(\Lambda, \Omega)$ and $S_\text{fin}(\mathcal{O},\Lambda)$ impossible for nontrivial spaces?
Recall that an open cover $\mathcal{U}$ of $X$ is a $\gamma$-cover if it is infinite and each $x\in X$ belongs to all but finitely many elements of $\mathcal{U}$ and an open open cover $\mathcal{V}$ ...
5
votes
0
answers
154
views
Does "achieving more GH-distances than some compact space" imply compactness?
Previously asked and bountied at MSE:
For complete metric spaces $X,Y$, write $X\trianglelefteq Y$ iff for every complete metric space $Z$ such that the Gromov-Hausdorff distance between $X$ and $Z$ ...
24
votes
2
answers
740
views
"All retracts are closed" and "all compacts are closed"
I want to follow the discussion from here concerning about the strength of the separation "all retract subspaces are closed".
(A retract subspace of a topological space $X$ is a subspace $A$ ...
5
votes
1
answer
197
views
How many pairwise non-homeomorphic non-empty closed subsets of the Cantor set are there? [duplicate]
My question is more or less related to basic set theory. But I don't know even that. Apologies if I added the wrong tags.
Motivation: How many non-compact (planar) surfaces are there upto ...
8
votes
0
answers
238
views
First order formula describing connected components
I ask this question here after no answer came up in the original MathSE question.
Let $\mathcal{L}$ be the language $\{+,-,\cdot,0,1,P\}$ where $P$ is some $n$-ary relation symbol. Is there a formula $...
6
votes
1
answer
269
views
Extending a partially defined metric on a metrizable space
Let $X$ be a metrizable topological space, $A\subseteq X\times X$ a nonempty closed subset which is reflexive, symmetric and transitive, $d:A\to \mathbb{R}_+$ a continuous function that satisfies the ...
2
votes
1
answer
178
views
A stronger version of paracompactness
Given a topological space $(X,\tau)$, recall that a cover $\mathcal{U}$ of $X$ is locally finite if for every point $x\in \mathcal{U}$ has a neighborhood $U$ that intersects finitely many elements of $...
1
vote
0
answers
104
views
Moore space over a group with infinite generator
I am not an expert on this topic. I am trying to learn about Moore spaces of type $(G,n)$. where $G$ is abelian and $n\geq 2$.
Let $M$ be a simply connected non-compact $4$-manifold with $H_2(M;\...
6
votes
1
answer
164
views
Subobject classifier in $\mathsf{Top}^{D^{\text{op}}}$?
Let $D$ be a small category. Does the category of diagrams $\mathsf{Top}^{D^{\text{op}}}$ have a classifier of (strong?) subobjects? I tried following the "sieve construction" for the ...
4
votes
2
answers
242
views
Curves in the plane and their number of holes
Suppose that the closed, piecewise $C^1$-curve $f(\mathbb T)$ has exactly $n$ points that are run through twice, all other points are run through once. Is it true that the compact set $f(\mathbb T)$...
8
votes
4
answers
597
views
Uniform density of Lipschitz maps is space of continuous function — for general metric spaces
Let $X$ and $Y$ be metric space, $X$ be compact, $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with uniform convergence on compacts topology, and $\operatorname{Lip}(X,Y)$ denote the ...
3
votes
1
answer
140
views
Spaces satisfying a strong Cartan-Hadamard theorem
Let $(X,d)$ be a connected geodesic metric space. When does there there exists a covering map $\pi:H\rightarrow X$ which is a local-isometry where $H$ is either a Hilbert space or a Euclidean space?
...
4
votes
1
answer
446
views
On the definition of a continuous function
I remember once reading that "a continuous function can be loosely described as a function whose graph can be drawn without lifting the pen from the paper". We all know that this is not true....
6
votes
0
answers
270
views
Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
1
vote
0
answers
87
views
Reference request: rates of weak convergence of Polish space-valued random variables
Let $(E,\mathscr{E})$ be a Polish space $E$ together with its Borel $\sigma$-algebra $\mathscr{E}$. Let $(\Omega, \mathscr{F},P)$ be a probability space and let $(X^{(n)})_{n \in \mathbb{N}}$ be a ...
2
votes
1
answer
200
views
A characterization of continuity in terms of preservation of connected sets. Where to find the result?
There is a result that if $X$ is a locally connected space and $Y$ is a locally compact Hausdorff space, then a function $f \colon X \to Y$ is continuous if and only if $f$ has a closed graph and for ...
3
votes
1
answer
195
views
Is it possible to characterize the contractible subsets of $\mathbb{R}^n$?
It is similar to "trees" of sets homeomorphic to star-shaped sets tangent to each other by a point (the edges correspond to tangency). Is that all, or are there contractible sets that don't ...
6
votes
0
answers
130
views
A theorem by R.L. Moore
The following result is due to R.L. Moore.
Let $K\subseteq\mathbb C$ be compact. Suppose that
$K$ is connected,
and that $\mathbb C\setminus K$ is connected.
Then $\partial K$ is connected.
Does ...
11
votes
2
answers
437
views
When is a k-space locally compact?
We're looking at the possible cardinal sequences of LCS (locally compact, Hausdorff, scattered) spaces, which has led us to think about taking a quotient of a locally compact, scattered space.
A k-...
3
votes
0
answers
156
views
Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?
I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
3
votes
0
answers
224
views
How rigorously can we apply the data supplied by this nonstandard attack on Kuratowski's closure-complement problem?
Suppose a student assigned an advanced version of Kuratowski’s closure-complement problem to solve—one that leaves out the standard hint about the finite upper bound of $14$—decides to look for the ...
2
votes
1
answer
338
views
$4$-manifold with simply connected boundary
This may be a very silly question but I could not get any counter-example.
Let $M$ be a compact differential $4$-manifold with boundary $dM$.
Suppose that the inclusion map induced map $\pi_1(dM) \to \...
2
votes
0
answers
157
views
Presentation complex and arbitrary $2$-dimensional CW-complex with same fundamental group
Given a finite group $G$, consider a presentation $P$ of $G$ and consider $X_P$, the presentation complex. Now let $Y$ be any $2$-dimensional CW-complex with $\pi_1(Y)=G$. Is there any relation ...
15
votes
1
answer
508
views
Can ($X^I$, product topology) and ($X^I$, box topology) be homeomorphic for some nontrivial $X$ and infinite $I$?
This is a verbatim repost of this question by Jianing Song. A few months ago I placed a bounty on the question but there were no answers, so I am reposting it here.
Let $X$ be a nontrivial ...
3
votes
1
answer
313
views
Extremely disconnected or extremally disconnected?
In the context of Banach space theory, what is the correct terminology: extremally disconnected or extremely disconnected. Looking through the internet I have met using both extremely and extremally ...
4
votes
1
answer
146
views
When does the refinement of a paracompact topology remain paracompact?
Let $(X,\tau)$ be a Hausdorff paracompact space. Let $\tau'$ be the smallest $P$-topology refining $(X,\tau)$, i.e. the topology which has for base the $G_\delta$-subsets of $(X,\tau)$.
Is it true ...
4
votes
1
answer
284
views
Is there any example of a Lindelöf space that has no Menger dense subspaces?
A space $X$ is said to be Menger if for each sequence $(\mathcal{U}_n)$ of open covers of $X$, there is a sequence $(\mathcal{V}_n)$ such that $\mathcal{V}_n$ is a finite subcollection of $\mathcal{U}...
7
votes
1
answer
418
views
$\Sigma_*$-product is not $\sigma$-countably compact
In Arhangel'skii's book "Topological function spaces" there is a part where the author uses that, if $\kappa>\omega$ is a cardinal number, then the space $$\Sigma_*(\kappa):=\left\{x\in \...
1
vote
0
answers
178
views
Is the domain space in Lusin's theorem required to be Hausdorff?
I'm reading a general version of Lusin's theorem, i.e.,
If $\mu$ is a finite Radon measure on $X$, and $Y$ is a second countable topological spaces, then for any Borel-measurable function $f:X\to Y$ ...
4
votes
0
answers
71
views
Is each typical map on the $n$-cube strongly rigid?
This question is inspired by this (still unanswered) MO-post.
A function $f:X\to Y$ between topological spaces is called strongly rigid if every continuous self-map $h:X\to X$ with $f\circ h=f$ is the ...
1
vote
0
answers
146
views
Second homology group of a presentation complex
I am trying to learn results related to the presentation complex of a group and I am new to this subject. So I apologize if the questions are silly.
Given a finite group $G$, and a presentation $P$ of ...
2
votes
1
answer
234
views
Formula for the Euler characteristic of a local system on $\mathbb{P}^1$
Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.
Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...
7
votes
1
answer
260
views
Are closed embeddings characterized by a left lifting property in the category of topological spaces?
It is well-known and easy to check that a continuous map between topological spaces is an embedding if and only if it has the LLP with respect to $A \to *$ and $B \to *$ where $A$ is the two-point ...
4
votes
1
answer
146
views
Given $f$ from the cylinder $C$ to the interval constant on one boundary, is there a $r:C\to C$ constant on a boundary with $f\circ r = f$?
My question might be trivial, but my lack of knowledge of this particular subject has not enabled me to find the answer. What I want to know is the following. Let $I=[0,1]$ and $C=S^1\times I$ be the ...
1
vote
1
answer
192
views
Name of a space with both a topology and a metric that are not compatible?
Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$.
Is there a canonical name for such a structure (maybe ...