Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,447
questions
2
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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\}|\...
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2
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1
answer
65
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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: \...
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-4
votes
1
answer
162
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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,...
1
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0
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46
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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 ...
- 159
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56
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Geometric intuition behind hyper-sphere volume recurrence relation [closed]
There is a recurrence relation for calculating the volume of a hyper-sphere and a logical explanation for why it holds, as well as other explanations here on MO. Is there a geometric intuition behind ...
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2
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0
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71
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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 ...
2
votes
0
answers
44
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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 $\...
- 463
2
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0
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111
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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=\{\...
- 1,045
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2
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177
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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 ...
- 61.6k
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1
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216
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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 ...
- 197
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1
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106
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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 ...
- 345
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1
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76
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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 ...
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+100
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{...
- 4,969
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2
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272
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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-...
- 1,045
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3
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224
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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 ...
-1
votes
1
answer
83
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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 ...
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80
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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)=...
- 31
12
votes
1
answer
301
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 ...
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-2
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0
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452
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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 ...
- 308
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0
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110
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Extremally disconnected sets as building blocks for compact Hausdorff spaces
Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
- 1,446
-1
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0
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115
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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.
1
vote
1
answer
79
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 ...
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124
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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 $\...
- 755
2
votes
0
answers
72
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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.
...
5
votes
1
answer
227
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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 $B/A$. ...
- 573
1
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1
answer
77
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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 $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
- 107
1
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0
answers
70
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Uniform approximation over compacts using weighted function spaces
I'm interested in approximations over the so-called weighted function spaces.
Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
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7
votes
1
answer
340
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Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
Prove the inequality
$$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$
where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
- 79
5
votes
1
answer
254
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Quotients in categories of metric spaces
There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
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0
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142
views
Connectedness of deleted symmetric product
Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
- 622
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385
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$$ \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(...
- 625
8
votes
1
answer
405
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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 $\sum a_n$ ...
- 3,104
6
votes
1
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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 ?
- 1,071
2
votes
0
answers
24
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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 ...
- 81
1
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0
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61
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 ...
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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 ...
2
votes
1
answer
161
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Factorization systems for vector bundles
Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
- 47
1
vote
1
answer
110
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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 ...
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3
votes
1
answer
171
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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\...
- 1,071
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votes
0
answers
82
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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) ...
0
votes
0
answers
49
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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 ...
- 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, \, \...
- 419
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votes
0
answers
79
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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\...
- 56.6k
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 ...
- 21
0
votes
0
answers
58
views
Computing the eta invariant of a rather contrived operator on the circle
For physical reasons, I am interested in computing the eta invariant of the following Hermitian operator acting on complex valued functions on the circle with circumference 1. I define the operator ...
- 111
1
vote
2
answers
120
views
Axiomatic definition of Katětov closure operator
In the book "Categorical Structure of Closure Operators with Applications to Topology" by Dikranjan and Tholen a Katětov closure operator is defined in terms of filter covergence:
$k_X(M):=\{...
11
votes
1
answer
280
views
Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?
In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$.
The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
- 56.6k
1
vote
1
answer
125
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Points in the Stone Cech compactification are intersection of open sets
Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...
3
votes
1
answer
576
views
What is the Lebesgue covering dimension of this topological space?
Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal.
Take the induced topology defined by the Lorentzian metric called Alexandrov topology.
This topology matches ...
- 308
15
votes
1
answer
751
views
What is this equivalence relation on topological spaces: there are bijective continuous maps in both directions
Consider the following equivalence relation on topological spaces:
$X\sim Y$ $:\Longleftrightarrow$ there are bijective continuous maps $\phi:X\to Y$ and $\psi:Y\to X$.
Note that there are no ...
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