All Questions
5,185 questions
5
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Is a circle of circles necessarily a 2-manifold?
Let $X$ be a continuum (a compact connected metric space).
Suppose that there is a collection of simple closed curves $\mathcal C$ which partitions $X$ into pairwise-disjoint, nowhere dense sets. ...
- 3,317
6
votes
2
answers
830
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Do all homogeneous spaces have homogeneous compactifications?
Let $X$ be a separable metric space which is homogeneous, i.e. for every two points $x,y\in X$ there is a homeomorphism $h$ of $X$ onto itself such that $h(x)=y$.
A compactification of $X$ is a ...
CommunityBot
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0
votes
0
answers
39
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Countably infinite monoids with minimal right ideals
Is there any classification of countably infinite monoids with minimal right ideal? or at least in some classes of monoids?
- 237
0
votes
0
answers
180
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Proof of Co-H map the map $f:\Sigma SU(4)\rightarrow \Sigma^2 \mathbb{CP^3}$
How to show the map $f:\Sigma SU(4)\rightarrow \Sigma^2 \mathbb{CP^3}$ is Co-H-map?
5
votes
2
answers
188
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$|\mathsf{RO}(X)|$ vs. $2^{d(X)}$ for $T_3$ spaces
Let $\mathsf{RO}(X)$ stand for the collection of regular open subsets of a topological space $X$ and let $d(X)$ be its density. It is well-known (see Theorem~3.3 of Hodel's chapter in the Handbook) ...
- 1,364
5
votes
1
answer
254
views
Is the topology of weak+Hausdorff convergence Polish?
Let $X$ be a compact metric space, $P_X$ the set of Borel probability measures on $X$, and $K_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff ...
- 42k
3
votes
2
answers
301
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Upper density of subsets of an amenable group
Let $G$ be an amenable group (so locally compact Hausdorff) and also assume it is second countable if needed. My question is that what are the standard ways (across literature) of defining the upper ...
- 63.9k
6
votes
1
answer
484
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Why finitely presentable objects in Top need to be discrete?
In Locally Presentable and Accessible Categories, page 12 (10),
A topological space is finitely presentable in $\mathbf{Top}$, the category of topological spaces and continuous functions, iff it is ...
- 28.2k
0
votes
1
answer
101
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A question on relation of different triangulations of a triangulable space
Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, i.e. the restriction on the boundary is itself a triangulation, is it these ...
- 5,544
41
votes
4
answers
2k
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What is the probability two random maps on n symbols commute?
It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...
- 7,545
3
votes
1
answer
315
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Kuratowski-Ulam Theorem, nowhere dense set in product space
Let $E$ be a nowhere dense subset of $\mathbb{R}\times \mathbb{R}$.
For $x\in \mathbb{R}$, define
$$E_x=\{ y\in\mathbb{R}\mid (x,y)\in E\}.$$
Let $D$ denote the set of $x$ for which $E_x$ is NOT ...
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26
votes
1
answer
1k
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Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
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2
votes
1
answer
119
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Density of smooth bi-Lipschitz maps in smooth maps
Setup/Motivation:
Let $(M,g)$ and $(N,\rho)$ be complete Riemannian manifolds of respective dimensions $m$ and $n$ and suppose that $m\leq n$. Let $\operatorname{bi-C}^{\infty}(M,N)$ denote the class ...
- 26.3k
1
vote
0
answers
84
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Terminology for the property: "Each uncountable disjoint open family is locally countable"
Suppose that a topological space $X$ satisfies the following property
(P): "Each uncountable disjoint open family is locally countable",
where a family $\mathcal U$ of subsets of $X$ is ...
- 63.9k
0
votes
10
answers
9k
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What is an explicit example of a sequence converging to two different points? [closed]
In principle a sequence in a non-Hausdorff space can converge to two points simultaneously.
Can anyone give me an explicit example of the above?
Or tell me any method of generating such kinds of ...
- 4,713
4
votes
2
answers
336
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If $\Omega$ is locally Lipschitz, then $\Omega = \bigcup_{k = 1}^N \Omega_k$ for $\Omega_k$ star shaped with respect to an open ball $B_k$
I am reading Galdi's Introduction to the mathematical theory of Navier Stokes equations and there is an argument which comes up quite often that I really don't understand.
In many theorems of Chapter $...
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3
votes
0
answers
153
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Is it possible to reconstruct the universally measurable sets in X from the $C^*$-algebra $C(X)^{**}$?
This continues my question of two months ago. Let $X$ be a compact Hausdorff topological space. We consider the $C^*$-algebra $C(X)$ of continuous functions on $X$, its dual space $C(X)^{*}=M(X)$ of ...
- 7,426
7
votes
1
answer
271
views
Algebraic proof that the monoid ring of a torsion-free monoid is reduced
In what follows, I say that a monoid $M$ is torsion-free if the $n$-th power map is injective for all $n \geq 1$. I have a proof of the following result:
Claim: if $M$ is a torsion-free commutative ...
- 15.5k
3
votes
0
answers
126
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A path with zero increments and positive area
I am studying rough paths from the 2007 St Flour lecture notes and I came across the example at the end of chapter one of the sequence of paths $X(n):[0,2\pi]\to \mathbb R^2$ given by $X_t(n) = \frac{...
- 177
0
votes
1
answer
137
views
Examples of b-connected sets?
B is a b-open set if $B\subset Cl(IntB) \cup Int(ClB)$
A topological space $X$ is b-disconnected if it can be expressed as a union of two disjoint non-empty b-open sets. Otherwise, $X$ is said to be ...
- 1,364
8
votes
1
answer
491
views
The class of spaces where every Borel measure is atomic
I have been considering the following question:
Let $X$ be a compact, metrizable space with the following property: every (regular) Borel probability measure on $X$ is atomic, i.e. for each $\mu\in\...
2
votes
2
answers
134
views
On a generating set of numerical semigroups of multiplicity three
Let $S$ be a numerical semigroup. Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the Frobenius number of $S$; let $g(S)=|\mathbb ...
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5
votes
1
answer
134
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Decomposing $\{0,1\}^\omega$ endowed with the Sierpinski topology
Consider the Sierpinski space $\text{S} = (\{0,1\}, \tau)$ where $\tau = \big\{\emptyset,\{0\}, \{0,1\}\big\}$. Endow $\text{S}^\omega$ with the product topology.
If $X, Y$ are topological spaces with ...
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0
votes
0
answers
98
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Reference request: subspace-based generalisation of weak* convergence
Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
- 463
2
votes
1
answer
148
views
Borel $\sigma$-algebras on paths of bounded variation
Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$.
Let further $B\subset C$ be the subspace of $0$-started ...
- 463
1
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0
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97
views
Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
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0
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113
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Finite sets are residual in the Hausdorff space
Let $X$ be a metric space, let $\mathbb{H}(X)$ denote the set of non-empty closed subsets of $X$ with Hausdorff metric which we denote by $d_{\mathbb{H}(X)}$, and let $\mathbb{H}_{\operatorname{fin}}(...
- 5,405
2
votes
0
answers
75
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Dual space induced by a finer topology
Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two seminorms on a space $E$ such that $\|\cdot\|_2\geq\|\cdot\|_1$. Let further $E_i:=(E,\|\cdot\|_i)$ and
$$C_b(E_i):=\{f : E\rightarrow\mathbb{R}\mid f \ \...
- 463
7
votes
1
answer
353
views
Does the category of cosheaves have enough projectives?
Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell ...
- 30.3k
2
votes
0
answers
61
views
Space of continuous paths up to strict reparametrization
Take a Hausdorff topological space $X$. Take two distinct points $x$ and $y$ of $X$. Consider a set $U$ of continuous paths $p$ from $[0,1]$ to $X$ equipped with the compact-open topology such that: $...
- 3,901
2
votes
0
answers
101
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Concrete topological objects and notions in the category of locales
I have read Peter Johnstone's “The Point of Pointless Topology” and the idea that topological spaces are not quite the right abstraction for topology seems, at least philosophically, rather appealing. ...
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1
vote
1
answer
258
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Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?
I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below).
It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be
written as an ...
- 10.6k
7
votes
1
answer
280
views
Extending a finite Baire measure to a regular Borel measure
Let $X$ be a Hausdorff compact space, and let $\mathrm {Ba}$, $\mathrm {Bo}$ be its Baire, respectively, Borel, $\sigma$-algebras. Let $\mu:\mathrm {Ba}\to[0,+\infty)$ be a finite Baire measure: it is ...
- 60.6k
4
votes
2
answers
228
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Existence of a function on the Euclidean space which differs by constants from locally defined functions
Let $\{U_\lambda\}_{\lambda\in\Lambda}$ be an open covering of $\mathbb{R}^n$.
Given a family of functions $f_\lambda:U_\lambda\rightarrow \mathbb{R}\,(\lambda\in\Lambda)$ such that $f_\lambda-f_\mu: ...
9
votes
1
answer
588
views
How to prove the product of Whitehead manifold and $\mathbb{R}$ is homeomorphic to $\mathbb{R}^4$?
I am currently reading Rolfsen's "Knots and Links". At page 82 Whitehead manifold $W$ is defined and an exercise asking to show that $W\times \mathbb{R}\cong \mathbb{R}^4$ is left. Reference ...
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0
votes
1
answer
504
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Is every compact contractible subset of $\mathbb{R}^n$ homeomorphic to a closed ball of some dimension? [closed]
Question: Is every compact contractible subset of $\mathbb{R}^n$ homeomorphic to a closed ball of some dimension?
This post doesn't quite answer my question because it is about open sets.
- 6,995
3
votes
1
answer
255
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What is this property of surjective continuous maps called?
Let $f\colon X\to Y$ be a continuous map between topological spaces, which you can assume to be Hausdorff if you like. Say that $f$ has property $P$ if for every compact subset $L\subseteq Y$, there ...
- 44.4k
3
votes
2
answers
362
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Topological spaces with nowhere locally constant functions to the reals
I would like a nice characterization of topological spaces with continuous functions to the reals which are nowhere locally constant, i.e. not constant on any (non-empty) open set. For sure, the ...
- 42k
2
votes
0
answers
130
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Attaching a 2-handle to a once-twisted unlink in the boundary of the 4-ball
Consider the 3-sphere $S_3$ with an unlink loop $L$ whose tubular neighborhood is identified with the solid torus $B_2\times S_1$ with one twist, i.e., such that the image of $x\times S_1$ (where $x$ ...
- 3,001
35
votes
2
answers
5k
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Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"?
This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...
- 30.3k
1
vote
1
answer
183
views
Topological analog of the Lusin-N property
$A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets.
Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\...
- 1,848
2
votes
1
answer
130
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Gluing isotopic smoothings
Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
- 5,849
8
votes
1
answer
504
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On homeomorphic subsets of $\mathbb{R}^3$ with non-homeomorphic complements
Let $A,B$ be two homeomorphic topological subspaces of $\mathbb{R}^3$ such that their complements $\mathbb{R}^3 - A, \mathbb{R}^3 - B$ are not homeomorphic to each other. Must $A \cong B$ contain a ...
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1
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1
answer
193
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Is the image of a constructible set between Jacobson spaces constructible if the map takes closed points to closed points?
Let $X$ and $Y$ be two spectral Jacobson spaces and let $f: X \to Y$ be a spectral morphism, i.e. $f$ is continuous and the inverse image of a quasi-compact open is quasi-compact. Suppose further that ...
- 28.7k
4
votes
1
answer
280
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Product topology from two premetric spaces induced by sum of premetrics?
For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$.
Do ...
- 42k
2
votes
0
answers
82
views
Is every first-countable symmetrizable space perfect?
Let us recall that a symmetric on a set $X$ is any function $d:X\times X\to[0,\infty)$ such that
for every $x,y\in X$ the following two conditions are satisfied:
$d(x,y)=0$ if and only if $x=y$;
$d(...
- 42k
9
votes
0
answers
267
views
Can "$\exists\mathcal{X}(R\cong C(\mathcal{X}))$" be expressed in "large" infinitary second-order logic?
Originally asked and bountied at MSE without success:
Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ...
CommunityBot
- 1
3
votes
0
answers
80
views
Is $\mathfrak q_0$ equal to the smallest cardinality of a second-countable $T_1$-space which is not a $Q$-space?
A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$.
The smallest cardinality of a metrizable separable space which is not a $Q$-space is denoted by $\mathfrak q_0$ and ...
- 42k
8
votes
1
answer
709
views
Is every second-countable Hausdorff space symmetrizable?
Let us recall that a symmetric on a set $X$ is any function $d:X\times X\to[0,\infty)$ such that
for every $x,y\in X$ the following two conditions are satisfied:
$\bullet$ $d(x,y)=0$ if and only if $x=...
- 42k
4
votes
1
answer
194
views
Is every first-countable Lindelof space of cardinality $<\mathfrak c$ a $Q$-space under MA?
Definition. A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$.
It is clear that every $Q$-space has countable pseudocharacter (= all singletons are $G_\delta$) and is ...
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