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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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Smooth functions with certain properties on the sphere

Let $ f $ be a smooth function defined on the sphere such that the set of points where $ f(x) - f(\tilde{x}_y) $ vanishes divides $\mathbb{S}^2$ into exactly four regions for all $y\in \mathbb{S}^2$, ...
A random mathematician's user avatar
8 votes
0 answers
167 views

Big list of Hochster dual concepts

Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and ...
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Any useful bases for the topology induced by the $t$-Wasserstein distance?

I am working on $\mathbb R ^d$ equipped with the usual Euclidean metric. I know of one nice base for $\mathcal W _t$, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
uniform_on_compacts's user avatar
6 votes
2 answers
200 views

Reference: If $X$ is metrizable, then $X$ is realcompact iff $|X|$ is non-measurable

Note: What I call a measurable cardinal seems to be non-standard among set theorists, and should be called a $\sigma$-measurable cardinal. I know that a discrete space is realcompact iff its non-...
Jakobian's user avatar
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1 vote
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Included excluded point topology

I was introduced to included and excluded point topology fairly recently by my classmates and to my amazement the topologies mentioned worked for me in giving counter examples to many statements. I ...
Amrit Awasthi's user avatar
3 votes
2 answers
479 views

For every sequence of nonempty open sets there is a disjoint sequence of nonempty open sets "below" it

I am looking for any information about the following property for a compact Hausdorff space $K$: For any sequence $\left(U_{n}\right)$ of nonempty open sets (not necessarily distinct) there is a ...
erz's user avatar
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1 answer
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Local embedding and disk in domain perturbation

Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
monoidaltransform's user avatar
4 votes
1 answer
133 views

What is the extent of a $\Sigma$-product of a (uncountable) power of a (countable) discrete space?

Recall that a $\Sigma$-product of a family of spaces $\{X_s:s\in S\}$ with a base point $a=(a_s)\in \prod_{s\in S} X_s$ is the subspace $$\Sigma(a)=\{x\in \prod_{s\in S} X_s: |\{s\in S:a_s\neq x_s\}|\...
J. Casas's user avatar
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Hausdorff-Lipschitz continuity of cone correspondence

Let $\mathbb{R}_+$ denote the strictly positive real numbers, let $\mathcal{X} \subset \mathbb{R}^n$ and $\mathcal{P} \subset \mathbb{R}^m$ be compact and convex subsets, let \begin{equation} f: \...
Heinrich A's user avatar
-4 votes
1 answer
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Can a Polish space have two different topologies?

Let $X$ be a Polish space with the compatible metric being $d_1$. So $(X,d_1)$ is a separable complete metric space, and the topology is generated by $d_1$. Can there be a metric $d_2$ such that $(X,...
uniform_on_compacts's user avatar
1 vote
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Necessary or sufficient conditions for the $k$-fold intersection to be empty in a covering with a "tree structure"

Consider a finite collection of $d$-dimensional balls $\mathfrak{B}=\{B_1,\ldots,B_n\}$ which cover a PL $d$-manifold $M$, i.e. $M=\bigcup_{i=1}^{n}B_i$. Suppose we want to compute the Euler ...
rab's user avatar
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2 votes
0 answers
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Prokhorov's theorem for countably many random measures on a Polish space

I am looking for help to show the following lemma: Lemma Let $(\Omega,\mathcal A,\mathbb P)$ be a complete, standard Borel probability space and $\mathcal X$ a Polish space. Let $\mathcal P(\mathcal ...
Pierre-François Massiani's user avatar
2 votes
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Fundamental group of cyclic branched cover of affine plane

Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
Doug Liu's user avatar
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What is the universal/fine uniformity on a topological group?

Cross posted from https://math.stackexchange.com/questions/4889335 I'm aware that every topological group is uniformizable: given a neighborhood $U\in\mathcal N(e)$ of the identity, the set $D_U=\{\...
Steven Clontz's user avatar
7 votes
2 answers
180 views

Cover the $n$-disc irredundantly with $n+1$ open sets. Suppose that the $(n+1)$-fold intersection is empty. Then is some $n$-fold intersection empty?

Cover the $n$-disc with $n+1$ open sets $D^n = U_0 \cup \dotsb \cup U_n$. Suppose that $U_0 \cap \dotsb \cap U_n = \emptyset$. Suppose moreover that the cover is irredundant in the sense that no ...
Tim Campion's user avatar
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1 answer
217 views

Relationship between quotient CW-complexes after attaching cells

I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
William Thomas's user avatar
0 votes
1 answer
112 views

Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
MathLearner's user avatar
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1 answer
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Continuous modification of tangent vector fields

Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
MathLearner's user avatar
2 votes
0 answers
170 views

Homeomorphically deforming one continuous function to another

Motivation: A result of Klee Theorem 3.4 in [1], namely Klee's trick, implies that if $f,g:\mathbb{R}^d\to \mathbb{R}^n$ for any $d,n\in \mathbb{N}_+$ then there is a homeomorphism $\tilde{H}:\mathbb{...
ABIM's user avatar
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6 votes
2 answers
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Embeds in a topological W-group, or a W-space that embeds in a topological group?

In Theorem 3.11 of Tkachuk - A compact space $K$ is Corson compact if and only if $C_p(K)$ has a dense lc-scattered subspace it's shown that if a compact Hausdorff space embeds in a topological W-...
Steven Clontz's user avatar
0 votes
3 answers
224 views

Extending $\mathbb{R}$ to a higher dimensional manifold [closed]

If a topological space $X$ is Hausdorff, connected, second countable, homogeneous (i.e. it has transitive homeomorphism group) and embeds the real line $\mathbb{R}$, does it follow that $X$ is a ...
Pavlos Kazakopoulos's user avatar
-1 votes
1 answer
83 views

Is this function on the Cantor set continuous? [closed]

Let $S = \displaystyle \prod_{n \ge 1} \{ 0, 1\}$ be the set of binary sequences, so $S$ with the product topology is homeomorphic to the Cantor set. Endow $\mathbb{Z}_{\ge 1} \cup \{ \infty \}$ with ...
Joshua Ruiter's user avatar
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0 answers
80 views

Generator of an analytic semigroup

Perhaps I have a naive question. My question is as follows: When we consider a Cauchy proposition of the following form: $$ \begin{cases} x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\ x(0)=...
Jaouad's user avatar
  • 31
12 votes
1 answer
302 views

Approximate classifying space by boundaryless manifolds?

As pointed out by Achim Krause, any finite CW complex is homotopy equivalent to a manifold with boundary (by embedding into $\mathbb R^n$ and thickening), and so every finite type CW complex can be ...
0207's user avatar
  • 123
-2 votes
0 answers
456 views

The Lebesgue covering dimension of the Cosmic String interval topology

Take the spacetime $(M,g)$ that satisfies Einstein's Field Equations exactly where $g$ is locally: $$g= - c^2 dt^2 + d \rho^2 + (\kappa^2 \rho^2 - a^2) d \phi^2 - 2 ac d\phi dt + dz^2 \ $$ in the ...
Bastam Tajik's user avatar
1 vote
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110 views

Extremally disconnected sets as building blocks for compact Hausdorff spaces

Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
Peter Kropholler's user avatar
-1 votes
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115 views

Prove a generalization of hairy-ball theorem

I found an interesting question below: Prove that the 6-sphere admits no continuous field of tangent 3-planes.
threeautumn's user avatar
1 vote
1 answer
81 views

Real exponentiation in the quotients of rings of continuous functions by prime ideals

Consider the ring $C = C(X) = C(X; \mathbb{R})$ of continuous functions $f:X\to \mathbb{R}$ where $X$ is a Tychonoff space. This is naturally a lattice ordered ring by setting $f\geq 0$ iff $f(x)\geq ...
Jakobian's user avatar
  • 817
7 votes
0 answers
125 views

Is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?

In short, the question is in the title: is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis? A bit of context: Given a topological space $X$, a family $\...
Cla's user avatar
  • 755
2 votes
0 answers
72 views

Union of two open, open-unicoherent sets whose intersection is connected

I stumbled upon the following proposition, and haven't found an error in my proof yet. By "open-unicoherence" I mean unicoherence with closed sets replaced with open sets in the definition. ...
Calvin Wooyoung Chin's user avatar
5 votes
1 answer
227 views

In a topological group, is $G/A\to G/B$ a covering map if $A$ is open in $B$?

Let $G$ be a (Hausdorff) topological group, let $A,B$ be closed subgroups of $G$ such that $A$ is an open subgroup in $B$. Then we have an open continuous map $f:G/A\to G/B$, with typical fiber $B/A$. ...
Linus's user avatar
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1 vote
1 answer
77 views

Simple convergence of convex compact set implies Hausdorff convergence

I am wondering about the following : In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
Anthony's user avatar
  • 125
1 vote
0 answers
70 views

Uniform approximation over compacts using weighted function spaces

I'm interested in approximations over the so-called weighted function spaces. Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
Gaspar's user avatar
  • 81
7 votes
1 answer
345 views

Proving the inequality involving Hausdorff distance and Wasserstein infinity distance

Prove the inequality $$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$ where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
Luna Belle's user avatar
5 votes
1 answer
259 views

Quotients in categories of metric spaces

There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
Jochen Wengenroth's user avatar
0 votes
0 answers
142 views

Connectedness of deleted symmetric product

Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
Peluso's user avatar
  • 622
2 votes
0 answers
387 views

$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]

If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of $$ \left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
Ola Sande's user avatar
  • 625
8 votes
1 answer
409 views

The cardinality of projections of subsets of the Hilbert cube by inner products

I have three related questions. Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
Boaz Tsaban's user avatar
  • 3,104
6 votes
1 answer
139 views

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
Alexander Osipov's user avatar
2 votes
0 answers
24 views

Continuity of Kernel Mean Embeddings

Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
Gaspar's user avatar
  • 81
1 vote
0 answers
62 views

"Star" of a CW-complex

Suppose we have a CW-complex $X$ with a 0-cell $e^0$. Is the union of all the cells (of higher dimensions) for which $e^0$ is a boundary point open in $X$? I don't know if it has a name, but a similar ...
brattok's user avatar
  • 11
0 votes
1 answer
48 views

Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K? [closed]

Let $S$ be a compact connected surface. Let $K$ be a compact subset of $S$ and suppose that $K$ has a finite number of connected components. Let $U$ be a connected component of $S \setminus K$ and ...
Fernando Oliveira's user avatar
2 votes
1 answer
161 views

Factorization systems for vector bundles

Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
Nash's user avatar
  • 47
1 vote
1 answer
110 views

Variants of Dirichlet-type function as a pointwise limit of continuous functions

Problem Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both ...
hmeng's user avatar
  • 129
3 votes
1 answer
171 views

Is there a metric separable space with the following properties...?

Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$. Is there a metric separable space $X$ with the following properties: $|X|\geq\...
Alexander Osipov's user avatar
0 votes
0 answers
82 views

Finding an example if it exists, for a non-contractible and contractible space with special requirement on quotients of their union?

Let $A$ and $B$ be subsets of $n$-dimensional Euclidean space $\mathbb{R}^{n}$, such that $A$ is non-contractible, $B$ is contractible and $B$ is not an one-point set. I would like to find example(s) ...
Himanshu Yadav's user avatar
0 votes
0 answers
49 views

Approximating open subset of profinite group by union of cosets of ideal

I am trying to understand the proof of Theorem 1.3 in this paper by poonen. Poonen refers to Lemma 20 in a different paper. He claims that the open subset $U_P \subseteq \hat{\mathcal{O}}_P$ can be ...
jb1403's user avatar
  • 1
3 votes
0 answers
93 views

Finitely generated Banach lattice $C(K)$ and partitions of unity

Let $E$ be a Banach lattice. A Banach sublattice $L$ of $E$ is called finitely generated if there exists a finite subset $F \subseteq E$ such that $$L = \bigcap \{ \hat{L} \mid F \subseteq \hat L, \, \...
Julian Hölz's user avatar
5 votes
0 answers
79 views

When a compact subset of a TVS can be continuously projected on a closed linear subspace?

Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact. (Q): When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...
Pietro Majer's user avatar
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0 votes
0 answers
35 views

Estimate the gradient (with respect to local coordinates) of a partition of unity on a manifold

Suppose $\{U_\alpha\}$ is an atlas of coordinate patches of a (noncompact) smooth manifold $M$ of dimension $n$, with coordinates $(x_\alpha^1,\dots,x_\alpha^n)$ on $U_\alpha$. Furthermore we assume ...
Anar C's user avatar
  • 21

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