All Questions
5,183 questions
66
votes
4
answers
6k
views
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
- 103
9
votes
1
answer
889
views
Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory
In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
- 12.9k
6
votes
5
answers
953
views
Two arcs in the complement of a disc must intersect?
Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$.
Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...
- 4,786
7
votes
1
answer
669
views
Can $f: \mathbb{R}^2 \to \mathbb{R}$ be continuous, open and closed?
In the last few days I've been thinking on and off about these two problems and I can't get my head around them:
Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuous open map.
If $f$ is surjective ...
- 869
0
votes
2
answers
287
views
Distinguishable under manifold topology but indistinguishable under the Alexandrov topology
Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal.
In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
CommunityBot
- 1
7
votes
4
answers
3k
views
What is a good application of Urysohn's Theorem?
Urysohn's Metrization Theorem states that every Hausdorff second-countable regular space is metrizable.
What is an example of a Hausdorff second-countable regular space where it is difficult to prove ...
- 14.6k
1
vote
0
answers
101
views
When is the "Gelfand Remainder" compact?
Suppose we have a noncompact Hausdorff space $S$ and a Banach algebra $A \subset C^*(S,\mathbb R)$ of the space of real-valued bounded functions on $S$. For niceness let's assume $A$ separates the ...
- 1,955
0
votes
0
answers
72
views
Sequential compactness via Arzela-Ascoli theorem for uniform state spaces
Let $X$ be a uniform topological space and $C([0,1],X)$ the space of continuous functions from [0,1] to $X$. Assume that for subsets of $X$ sequential compactness and compactness are equivalent. Let $(...
- 133
0
votes
0
answers
42
views
Name for a sequence of open sets, each dense in the complement of the previous ones in the subspace topology
Let $X$ be a topological space. Let $\mathfrak{U} = \langle U_\alpha:\alpha\in\gamma\rangle$ be a sequence of non-empty open subsets of $X$ of length $\gamma$ ($\gamma$ an ordinal). Say (for now) that ...
- 5,429
7
votes
2
answers
448
views
Uncountable collections of distinct subsets of an interval (existence)
Throughout, $\mu$ is just the Lebesgue measure.
Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with $\mu(U_j) > 0$ for each $...
- 12.9k
11
votes
1
answer
961
views
Can the topologist's sine curve be realized as a Julia set?
Does there exist a rational function $f\in\Bbb{C}(z)$ whose Julia set coincides with
$$
T:=\left\{\left(x,\sin\left(\frac{1}{x}\right)\right)\,\Big|\,x\in\left(0,\frac{1}{\pi}\right]\right\}\cup\big(\{...
- 91.8k
5
votes
1
answer
350
views
Dévissage of stratified structures in Grothendieck's "Esquisse d’un programme"
I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “...
- 28.7k
2
votes
0
answers
71
views
Topological measure theory on spaces that are not completely regular
In the usual discourse regarding approaches to measure theory, it is often pointed out that the restriction of topological measure theory to locally compact Hausdorff spaces is insufficient. However, ...
- 3,816
5
votes
2
answers
3k
views
Zariski topology and compact \paracompact space?
Does the Zariski topology on a ring (not commutative in common) form a compact or paracompact space and why?
- 4,713
10
votes
0
answers
248
views
What is the tiling semigroup for an einstein "hat" tiling?
My undergraduate dissertation was on inverse semigroups and the key text I used for it was Lawson's, "Inverse Semigroups: The Theory of Partial Symmetries". In said book, Lawson describes ...
- 2,858
2
votes
1
answer
492
views
Scott topology, but for graphs
Would it be possible to define an order theoretic topology on graphs? I am thinking about the Scott topology. There would be an associated continuous map on graphs.
- 3,065
1
vote
0
answers
75
views
Trying to achieve "some sort of hemicompactness" in a Tychonoff space
Let $X$ be a Tychonoff space, i.e. Hausdorff and completely regular. Additionally, consider a map $\psi: X \to (0,\infty)$ such that $K_R := \psi^{-1}((0,R])$ is compact in $X$, for every $R>0$. ...
- 161
27
votes
1
answer
4k
views
connectivity of the group of orientation-preserving homeomorphisms of the sphere
In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...
- 1,446
5
votes
0
answers
96
views
$M^3$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not virtually nilpotent?
Let $M$ be a closed, orientable, irreducible 3-manifold and having an infinite fundamental group. Is it true that $M$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not ...
- 605
12
votes
1
answer
879
views
Partition of unity without AC
Several existence theorems for partition of unity are known. For example (source),
Proposition 3.1. If $(X,\tau)$ is a paracompact topological space,
then for every open cover $\{U_i \subset X\}_{i \...
- 11.4k
48
votes
19
answers
17k
views
What is your favorite proof of Tychonoff's Theorem?
Here is mine. It's taken from page 11 of "An Introduction To Abstract Harmonic Analysis", 1953, by Loomis:
https://archive.org/details/introductiontoab031610mbp
https://ia800309.us.archive....
- 6,353
2
votes
1
answer
162
views
A topological characterization of trees?
Motivated by this complex dynamics question:
Let $X$ be a compact, path-connected metric space. Suppose there exist an integer $N\geq 2$ and distinct points $p_1,\dots,p_N\in X$ such that no proper ...
- 23.7k
9
votes
2
answers
424
views
Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?
This question was firstly asked in mathematics stack exchange. Getting no answer, I copied it to here.
For a vector space $V$ over $\mathbb R$, I say a subset $S$ of $V$ is "anti-convex" if $...
- 10.6k
4
votes
0
answers
174
views
Centers and conjugacy classes of groups relative to a pair of group homomorphisms
$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by
\begin{align*}
\mathrm{Z}(G) &\...
- 11.8k
2
votes
0
answers
369
views
Constructing the Stone space of a distributive lattice
Does anyone have a good reference for the method of giving a topology to a distributive lattice as outlined in M.H. Stone's "Topological representation of distributive lattices and Brouwerian ...
- 4,219
33
votes
2
answers
2k
views
What happened to the last work Gaunce Lewis was doing when he died?
In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
- 21
1
vote
1
answer
248
views
Tightening a loop
Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
CommunityBot
- 1
2
votes
0
answers
81
views
Extension of a tangent vector field
Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
- 63.9k
2
votes
1
answer
185
views
Complete CCC Boolean algebras (or Stonean spaces)
I am interested in what is known about complete Boolean algebras $B$ with the countable chain condition (ccc), i.e., every disjoint set is countable. Let $K$ be the Stone space of $B$; the ...
- 11.4k
2
votes
1
answer
200
views
Subset in $[0,1]^k$ with positive density
Given a positive constant $0<\gamma<1$, does there exists integer $k_0>0$ such that for any integer $k\geq k_0$ the following holds?:
For any $A\subseteq\left[0,1\right]^k$ with the measure ...
CommunityBot
- 1
4
votes
0
answers
157
views
Existence of space $Z$ such that $\text{Cont}(X,Z) \cong X$
If $X, Y$ are topological spaces, let $\newcommand{\Cont}{\text{Cont}}\Cont(X,Y)$ denote the collection of continous maps $f:X\to Y$, and we endow $\Cont(X,Y)$ with the product topology inherited from ...
- 48.9k
4
votes
0
answers
108
views
Larger possible chain of closed subspaces in the dual of a Banach space
In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces.
My question is the following. If $X$ is an ...
- 153
2
votes
0
answers
73
views
Dual equivalence for multioperators
This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...
- 4,219
2
votes
1
answer
423
views
Conjecture about semigroups
Let $G$ be a finite semigroup with order $n$ odd. Let $S_i \in G, i=1,\ldots,\binom{n}{(n+1)/2}$ be all the subsets of $G$ of size $(n+1)/2$.
Let $E(S_i)$ be the set obtained "expanding" $...
- 14.6k
3
votes
1
answer
120
views
Non-isomorphic $T_0$-spaces with order-isomorphic topologies
Are there non-isomorphic $T_0$-spaces $(X_i, \tau_i)$ for $i = 1,2$ such that $\tau_1 \cong \tau_2$ when considered as partially ordered sets?
- 14.6k
2
votes
0
answers
156
views
Testing for weak homotopy equivalences with compact Hausdorff spaces
Let $f \colon X \to Y$ be a weak homotopy equivalence between topological spaces. If I am not mistaken, then one can rephrase this by stating that the induced map $[K,X] \to [K,Y]$ between homotopy ...
- 5,596
1
vote
2
answers
132
views
Description of atomless complete Boolean algebras with a countable $\pi$-base
Recall that a subset $A$ of a Boolean algebra $B$ is a $\pi$-base if for every $b>0$ there is $a\in A$ with $0<a\le b$. For example, the definition of atomicity says that atoms constitute a $\pi$...
- 11.4k
1
vote
0
answers
70
views
Is the property $S_1(\Omega,\Gamma)$ preserved when the metrizable separable topology is finer?
Let $(X,\tau)$ be a metrizable separable space and $X$ has the property $S_1(\Omega,\Gamma)$.
Suppose that $(X,\tau')$ be a metrizable separable space such that $\tau\subset \tau'$.
Will space $(X,\...
- 3,104
6
votes
0
answers
151
views
On dual notions of morphisms of algebraic structures obtained by replacing equaliser with coequalisers
This question is based on this discussion from the Category Theory Zulip. See also the earlier question Natural cotransformations and "dual" co/limits.
Let $G$ and $H$ be groups. We define ...
- 11.8k
6
votes
2
answers
308
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-...
- 11.4k
3
votes
2
answers
517
views
Several definitions of approximate continuity of real functions
I found the definition of approximate continuity stated as follows:
A function $f:\mathbb R \to \mathbb R$ is approximately continuous at $x_0$ iff there exists a set $A\in \mathcal L$ such that $x_0\...
- 1,309
7
votes
2
answers
2k
views
Every real-valued continuous function on a closed set of compact Hausdorff space has an extension.
I've noted, that the following fact can be proven in a few lines using $C^*$-algebra theory. I wonder if it has a simple elementary proof or not. Probably you can give me a reference.
Suppose $X$ ...
- 60.5k
5
votes
2
answers
406
views
Dimension of fibers under continuous maps
Is the following true? If yes, is there a simple way to show it?
Let $F:U \to \mathbb{R}^m$ be continuous, where $U$ is an open subset of $\mathbb{R}^n$. If $2 \leq m<n$, then there exists a fiber ...
- 2,154
1
vote
1
answer
104
views
Generalizations of Michael theorem
In [1] Michael proved the following:
Theorem. Let $f\colon X \to Y$ be continuous, closed, and onto, where $X$ is $T_1$. If $y \in Y$ is a q-point, then every continuous, real-valued function on $X$ ...
- 188k
8
votes
2
answers
3k
views
Connected components of the boundary of an open subset
Hi!
Let f be a (continuous, $C^\infty$... whatever) function from $\mathbb{R}^n$ ($n \geq 2$) to $\mathbb{R}$. Assume that each connected component of $f^{-1} (0; \infty)$ and $f^{-1} (-\infty; 0)$ ...
- 1,446
6
votes
1
answer
114
views
Filter vs Cover characterization of covering properties
In mathlib, topological properties are generally characterized in terms of filters wherever possible. In particular, a set $K$ is said to be compact provided that ...
- 18.6k
4
votes
1
answer
222
views
Is there an uncountable family of "hereditarily unembeddable" continua?
Define a family $\{C_i\}_{i\in I}$ of continua, that is compact connected metrizable spaces, to be hereditarily unembeddable (a name I just made up) iff for all $i\neq j$ no nontrivial subcontinuum of ...
- 770
6
votes
0
answers
197
views
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 ...
3
votes
1
answer
529
views
Zeros of a function defined on $\mathbb{S}^2 \times \mathbb{S}^2$
Let $u$ be a smooth function on the sphere, and for each $y \in \mathbb{S}^2$, let $R_y$ be the $180^\circ$ rotation about the vector $y$. For each pair $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...
- 3,599
3
votes
3
answers
1k
views
Countable atomless boolean algebra covered by a larger boolean algebra
Suppose $Q$ is an atomless countable boolean algebra, and $B$ is an arbitrary atomless boolean algebra. $Q$ is unique modulo isomorphisms. There is a subalgebra in $B$ that is isomorphic to $Q$. There ...
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