All Questions
5,183 questions
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Hereditary property of bionto (bi-onto) functions
Let $\,\ X\,\ Y\ $ be topological spaces. Set $\ A\subseteq X\ $ is said to be clopen in $\ X\ $ iff both $\ A\ $ and $\ X\setminus A\ $ are open. Continuous function $\ f:X\to Y\ $ is said to be ...
- 6,476
1
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0
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111
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Unique Hausdorff topology on trivial vector bundle?
Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
- 11
7
votes
1
answer
170
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Topological rigidity of cartesian product with $\mathbb{R}$
It seems that the following is true : if $V$ and $W$ are compact differentiable manifold of the same dimension, and $\mathbb{R} \times V$ is diffeomorphic to $\mathbb{R} \times W$, then $V$ and $W$ ...
- 12.3k
4
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0
answers
98
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Let $ G $ be a Lie group and $ H $ a connected subgroup of $ G $. If $ N_G(H)/H $ is finite does that imply $ H $ must be closed in $ G $?
Let $ G $ be a Lie group and $ H $ a connected subgroup of $ G $. If $ N_G(H)/H $ is finite does that imply $ H $ must be closed in $ G $?
The assumption that $ N_G(H)/H $ is finite cannot be weakened ...
- 4,081
0
votes
1
answer
145
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Can we describe open cover compactness of a space in how the space relates to other spaces?
I've seen two definitions of connectedness of categorical flavour which I present below:
(Maps into two point set): A topological space $X$ is connected iff the only continous functions $f:X \to \{ 0,...
6
votes
3
answers
264
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Can a scattered profinite set continuously surject onto a non-scattered profinite set?
A topological space is scattered if every nonempty subset has an isolated point. Are there any continuous surjections from a scattered profinite set to non-scattered profinite set?
- 2,356
3
votes
0
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145
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What is an example of a non-tight probability measure?
Billingsley (Convergence of Probability Measures, 1968) and van der Vaart and Wellner (Weak Convergence and Empirical Processes, 2023) discuss the concept of tight probability measures and use the ...
- 277
3
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0
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93
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Finiteness of non-orientable 3-manifolds with the same orientable two-fold cover
Given a compact, orientable and boundary incompressible 3-manifold $M$. Suppose that either $M$ is closed, or $\partial M$ consists of tori.
For which non-orientable 3-manifolds $N$, the orientable ...
- 605
7
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1
answer
134
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Universally closed implies proper for locales
It is well known that:
Theorem.
For a locale (resp. topological space) $X$, the following are equivalent:
$X$ is compact, i.e. every open cover of $X$ has a finite subcover.
For every locale (resp. ...
- 15.9k
1
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0
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262
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Is every self homeomorphism of the open disk conjugate to a homeomorphism extendable to the boundary?
Let $\mathbb{D}=\{z\in \mathbb{R}^2\mid |z|<1\}$
Is it true to say that every homeomorphism of $\mathbb{D}$ is conjugate to a self homeomrphism of the disk extendable to a homeomorphism of $\bar{\...
CommunityBot
- 1
1
vote
0
answers
37
views
Asymptotic growth of twisted alexander polynomials and hyperbolic volume for infinite families of knots
Let $\{K_n\}_{n=1}^\infty$ be an infinite family of hyperbolic knots with increasing crossing number, and let $\rho_n: \pi_1(S^3 \setminus K_n) \to SL_N(\mathbb{C})$ be a sequence of irreducible ...
60
votes
6
answers
7k
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Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
- 4,713
3
votes
0
answers
89
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Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 10.5k
6
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0
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98
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Rigid plane curves
A curve is a continuous one-to-one image of the real line $\mathbb R$.
A space $X$ is rigid if the only homeomorphism of $X$ onto itself is the identity.
Is there a rigid curve in the plane?
I am ...
- 3,317
6
votes
1
answer
360
views
On connected sum of compact manifolds along a submanifold
Let $M_1$ and $M_2$ be two compact manifolds of dimension $n\ge 3$. Let us have embeddings $i_1: K \to M_1$ and $i_2: K \to M_2$ for a closed manifold $K$ of dimension at most $n-1$ such that the ...
- 2,479
3
votes
1
answer
161
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Stone-Čech compactification of a Boolean subalgebra of $\{0,1\}^S$
Setup: Let $S$ be a set. Let $B$ be a Boolean subalgebra of $\{0,1\}^S$; i.e., just to be clear $B$ contains the constant $0$ and $1$ functions, and is stable under binary pointwise $\land$, $\lor$ ...
- 32.5k
9
votes
1
answer
456
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Topos notions coming from topology and uniqueness of generalizations
Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call ...
- 42.4k
15
votes
1
answer
601
views
Topological spaces in which countable intersections of dense open sets have dense interior
In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense.
Now consider the following strengthening of the Baire ...
CommunityBot
- 1
3
votes
1
answer
127
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Perfectly normal but not collectionwise normal space in ZFC
In the article A Perfectly Normal, Locally Compact, Noncollectionwise Normal Space Form $\lozenge^\ast$ by Daniels and Gruenhage (I presume "form" is a typo and it should be "from",...
- 5,596
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2
answers
489
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Does regular $G_\delta$ imply normal?
I'm trying to prove that if every closed set in a topological space is regular $G_\delta$, then the space is normal. By regular $G_\delta$, I mean for any closed set $A$,
there exists a countable ...
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51
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5
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9k
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Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 35.5k
2
votes
1
answer
49
views
Is any submetrizable linear topology linearly submetrizable?
Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$.
Is ...
- 16.4k
2
votes
0
answers
146
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Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R}$
About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the ...
- 356
3
votes
0
answers
90
views
Versions of the Fréchet–Urysohn property
Recall that a topological space is called Fréchet–Urysohn if every convergent net contains (as a set) a sequence, which is convergent to the same limit. I want to refine this property as follows.
Let $...
- 5,529
3
votes
1
answer
216
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Existence of a quasi-open (a.k.a semi-open) map into a Cantor cube
Recall that a topological space is extremally disconnected if the closure of any open set is open.
A continuous map is quasi-open if it maps nonempty open sets onto sets with nonempty interior. For ...
CommunityBot
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1
vote
0
answers
87
views
Convergence and sequential compactness for nonlinear operators
I have a family of operators $T_n\colon X \to Y$ where $X,Y$ are Hilbert spaces. These operators are nonlinear.
What kind of notions of convergence does one have for such operators? I'm specifically ...
- 2,830
4
votes
0
answers
148
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Isomorphism between the reduced C*-algebra of a groupoid and the crossed product of inverse semigroups
In Paterson's book "Groupoids, Inverse Semigroups and their Operator Algebras" he proves that for any r-discrete groupoid $G$ with unit space $G^0$, its full $C^* $-algebra $C^* (G)$ is ...
CommunityBot
- 1
5
votes
1
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371
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A problem of non-emptiness of intersections of certain chains of regular open sets
Let $X$ be a topological space and $\mathrm{RO}(X)$ its complete boolean algebra of regular opens. Define well inside relation: $$U\prec V\iff\overline{U}\subseteq V.$$ Let $\mathcal C\subseteq\mathrm{...
- 4,219
6
votes
1
answer
99
views
Continuous collections of arcs
Let $X$ be a separable metric space. Suppose there is a mapping $f:X\to C$ of $X$ onto the Cantor set $C$, whose point preimages are arcs (homeomorphic to $[0,1]$), and such that if $c_n\to c$ in $C$ ...
- 770
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0
answers
114
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Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
- 63.9k
0
votes
0
answers
98
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Does suspension preserve the inequivalence of knots?
Let $S$ be the suspension operator. Let $K1$ and $K_2$ be two knots in $S^3$ which are not equivalent. Does this imply that their suspensions in 4 sphere are not equivalent in the sense ...
- 12.9k
1
vote
1
answer
130
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A finite dimensional continuum with a subset $A$ such that both $A$ and $X\setminus A$ are dense and contractible
Inspired by this question we ask the following question.
Note that the comment conversation and answers to the above question imply that
There are two complementary subsets of the unit ...
- 356
8
votes
4
answers
526
views
Stone-Čech boundary is not extremally disconnected
Recall that a topological space is called extremally disconnected if the closure of every open subset is still open. Every discrete space is of course extremally disconnected, and the standard non-...
- 31.5k
3
votes
1
answer
165
views
Menger and Scheepers subsets of $\mathbb R$
$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...
- 505
0
votes
1
answer
129
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Ordering preserved by an inverse frame homomorphism
Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames):
Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$.
Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$.
...
- 4,219
3
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0
answers
161
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On generators of the multiplicative semigroup $\{r\in\mathbb Q:\ r>1\}$
The set $M=\{r\in\mathbb Q:\ r>1\}$ is a commutative semigroup with respect to the multiplication. For any integers $a>b\ge1$, we clearly have
$$\frac ab=\prod_{n=b}^{a-1}\frac{n+1}n.$$
So the ...
- 15.6k
4
votes
1
answer
223
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Existence of disintegrations for improper priors on locally-compact groups
In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
CommunityBot
- 1
3
votes
0
answers
92
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Reference for the monoidal category structure $X \otimes Y = X + Y + X \times Y$ on a distributive category
Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X ...
- 10.7k
1
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2
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220
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A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$
Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$.
Note that $A$ is non-empty with a Baire category argument.
I ...
- 869
3
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1
answer
226
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$\sigma$-product of the Hilbert cube
Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$
("eventually&...
CommunityBot
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3
votes
1
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161
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How to properly define a slice knot (or a locally flat disk)?
A knot $K\subset\Bbb S^3=\partial \Bbb D^4$ is said to be (topolopgically) slice if there is a locally flat disk $D\subset\Bbb D^4$ with $\partial D=D\cap \Bbb S^3=K$. As far as I understand, locally ...
- 2,797
4
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0
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154
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Is there a notion of "locally flat" for CW complexes?
A submanifold $X^n\subset Y^m$ is locally flat if each point $x\in X$ has a neighborhood $U\subset Y$ so that $(U,U\cap X)\simeq (\Bbb R^m, \Bbb R^n)$ with the standard embedding $\Bbb R^n\...
- 13.6k
6
votes
3
answers
1k
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Additional conditions under which separately continuous implies jointly continuous
Let $X,Y$ be compact metric spaces and consider $f:X\times Y\rightarrow X$ a separately continuous function.
I am wondering if there could be some additional conditions on $f$ (for example $f(\cdot,y):...
- 12.9k
13
votes
5
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5k
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Is the preimage of the closure the closure of the preimage under a quotient map?
Let $f : X \to X/\sim$ be a quotient map from a topological space $X$ to the quotient space $X/\sim$ for $\sim$ some equivalence relation. Let $S \subseteq X/\sim$. Is it true that $f^{-1}(\overline{S}...
- 109
3
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0
answers
129
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Topological interpretation of the existence part of the valuative criterion for properness
Let $X$ be a complex analytic space. I am trying to understand if there is a topological counterpart to the existence part of the valuative criterion for properness. The latter reads: every (ADDED: ...
- 283
5
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0
answers
185
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Stone–Weierstrass theorem for topological fields
It was showed in "The Stone–Weierstrass Theorem for valuable fields" that the Stone–Weierstrass theorem holds for any topological field whose topology comes from an absolute value or a Krull ...
- 12.9k
15
votes
3
answers
1k
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Classification of Platonic solids
My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula $v-e+...
- 4,713
7
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0
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349
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An open set which is not the union of a closed set and a countable set
The following fact is probably a known result:
Fact. Let $X$ be an uncountable Polish space. Then there exists an open subset of $X$ which is not the union of a closed set and a countable set.
Proof:...
- 12.9k
18
votes
2
answers
2k
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Which platonic solids can form a topological torus?
8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the ...
- 1,446
3
votes
1
answer
177
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Compactness of set of measurable functions between compact subspaces of real numbers
Let $X$ be a compact subset of $\mathbb{R}^n$ and $Y$ be a convex and compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find ...
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