All Questions
1,339 questions with no upvoted or accepted answers
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48
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Compute irreducibles of monoid
Given $n > 0$ and $w \in \mathbb{Z}^n$. Is there an efficient algorithm to compute the set of irreducible elements of the monoid $M_w = \{x \in \mathbb{N}^n \mid \langle x,w\rangle = 0 \}$?
Here, ...
2
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0
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290
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How to compute fundamental groups of slice disk complements?
To compute the fundamental group of the complement $S^3 \setminus K$ of a knot, one usually uses the Wirtinger algorithm. Is there a similarly well-established procedure for computing the fundamental ...
- 343
2
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60
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Are there finitely-presented astral monoids?
We say a semigroup $S$ is $k$-astral if there exists a finite set $F \subset S$ such that
whenever $s_1, s_2, ..., s_k \in S$ there exists $s \in S$ such that $\forall i: s_i \in sF$. Say $S$ is ...
- 6,652
2
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60
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Sobolev tensor spaces and finite ranks
Let $W^{2,2}(\Omega_i)$, $\Omega_i = [-1,1]$, $i = 1,\ldots,d$ be the Sobolev spaces of twice weakly differentiable, square integrable functions. Let further $\otimes_a$ denote the algebraic tensor ...
- 390
2
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56
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Is the space of $p$-geometric rough paths is Homeomorphic to Frechet Space
Let $\Omega G^p([0,T];\mathbb{R}^n)$ be a space of $p$-geometric rough paths with values in $\mathbb{R}^n$. Is $\Omega G^p([0,T];\mathbb{R}^n)$ homeomorphic to some Fr\'{e}chet space?
- 5,405
2
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0
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126
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Homeomorphic extension to totally disconnected sets
Dear Mathoverflow Community,
I am looking for a reference for the following topological fact:
Fact
Let $E$ and $F$ be two totally disconnected compact subsets of the plane (can assume perfect if ...
- 2,154
2
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0
answers
215
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A closed point in the closure of any point in the closure of any point of an irreducible scheme
Let $X$ be the underlying space of an irreducible scheme. In particular, $X$ is non-empty.
Note that the closure of any point is a closed irreducible subset. We say that a point has codimension $n$ ...
user137767
2
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0
answers
120
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Two small uncountable cardinals related to Q-sets
A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$.
Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-...
- 42k
2
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66
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Galois Covering induces new Cover $Ind_H ^G(Y)$
I have a question about the construction of the so called "induced cover" introduced in Tamas Szamuely's "Galois Groups and Fundamental Groups" (see page 84):
We consider a group $G$ which contains a ...
- 6,038
2
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81
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Is there a normal separable sequential $\aleph$-space with uncountable extent?
It is a classical fact from the undergraduate course of General Topology that under CH (more precisely, under $2^{\omega_1}>\mathfrak c$) every separable normal space has countable extent, i.e., ...
- 42k
2
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240
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3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators
During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...
- 207
2
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159
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What is the boundary of the set $\{ x : dist (x ,\partial \Omega) > \alpha \}$ for a domain $\Omega$?
Let $\Omega$ is a bounded open domain in $\mathbb R ^n$, and $\alpha \geq 0$ a real number, and consider the set $ E_\alpha = \{ x \in \Omega : \text{dist}(x , \partial \Omega) > \alpha\} $, which ...
- 747
2
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101
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A Baire space with meager projections
Question. Is there a Baire subspace $X$ of a Tychonoff power $M^\kappa$ of some separable metrizable space $M$ such that for any countable subset $A\subset \kappa$ the projection $$X_A=\{x{\...
- 42k
2
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77
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A possible characterization of stratifiable spaces?
Let us recall that a regular topological space is semi-stratifiable if each point $x\in X$ has a countable family of neighborhoods $(U_n(x))_{n\in\omega}$ such that each closed subset $F\subset X$ is ...
- 42k
2
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248
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Refining monotone-light factorizations
Let $f:X\to Y$ be a continuous map between topological spaces. Consider the quotient map $\pi:X\twoheadrightarrow X/E$ given by decomposing the fibers of $f$ to their connected components.
In Lemma 6....
- 10.5k
2
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69
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Invariant compact in division ring
Let $K$ be a discrete valued (with discrete valuation $v$) complete local division ring with ring of valuation $V$. Let $F$ be a compact subset of $V$. Suppose that for all $x\in F$ and for all $y\in ...
- 3,998
2
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62
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Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?
This is cross post to the question at MSE.
Let $V_1, V_2, \dots, V_n$ be a collection of vector subspaces in $\mathbb R^n$. For each $j=1, \dots, n$, $\dim(V_j) = m$ with $2 \le m < n$. We also ...
- 195
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72
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Topological Shape Operator More Sensitive than Inverse Limits
This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...
- 439
2
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49
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Maps having the right lifting property against cofibrations of compact spaces
I would like to know the properties of the maps that have the right lifting property against cofibrations of compact spaces. By definition, they are acyclic Serre fibrations, but I would hope to be ...
- 765
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333
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A property of subspaces of a topological space
Let $X$ be a topological space and $A$ be a subset of $X$. Is there any nice condition on $A$ equivalent to the property that if $C$ is a connected component of $X$, then either $A\cap C=\emptyset$ or ...
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158
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When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?
This is a cross-post to the question I asked at MSE.
Let $A \in M_4(\mathbb R)$ and $A = (e_2, x, e_4, y)$ where $e_2, e_4$ are standard basis in $\mathbb R^4$ and $x,y$ are undetermined variables. ...
- 195
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72
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Lifting clopen subsets
Let $A$ be a subset of a topological space $T$, we say that clopen subset of $A$ lift to $T$ whenever $L$ is a clopen subset of $A$ the there exists a clopen subset $H$ of $T$ such that $H\cap A=L$.
...
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102
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Square Peg Problem and curve density
Square Peg Problem (or conjecture) is so famous. See this article
Let $CS:=\{\gamma:S^1\longmapsto\mathbb{R}^2 | \;\;\text {Square Peg Problem is true}\}$ and $C=\{\Upsilon:S^1\longmapsto\mathbb{R}^2 ...
- 4,784
2
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82
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Enveloping a Jordan curve with a trace of another one
This question is inspired by this one, or rather the way I understood it.
Let $\gamma$ and $\delta$ be parametrised Jordan curves on the plane (i.e. homeomorphisms from $S^1$ onto its image in $\...
- 5,529
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48
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When $C_k(X,2)$ is Baire?
$C_k(X,2)$ is the family of all continuous function from $X$ to $\{0,1\}$ with compact-open topology.
for which kind of topological spaces $C_k(X,2)$ is Baire or meager?
Or is there something ...
- 123
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98
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Union of Two Faces, using the Jordan Curve Theorem
Consider four disjoint points in the plane, $v_{1}$, $v_{2}$, $v_{3}$ and $v_{4}$.
The cycle, $C:=v_1v_2v_{3}v_{4}v_{1}$, is the union of the (Jordan)
arcs, $A_{12}$, $A_{23}$, $A_{34}$, and $A_{41}$, ...
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108
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Homogeneity of the space of semicontinuous functions
I am interested in the topological homogeneity of function spaces.
Question. Let $X$ be a Tychonoff space, let $USC(X)$ be a space of upper semicontinuous functions on $X$ and let $USC(X)^+$ be a ...
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80
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Are the roots of an infinitely divisible probability infinitely divisible themselves?
Let $\mu$ be an infinitely divisible probability on a topological group $G$. If $\nu ^{* n} = \mu$ for some $n$, is $\nu$ an infinitely divisible probability too?
A sufficient criterion would be to ...
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81
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If $H$ and $Z$ are closed subgroups generating $G$, is $H \times Z \rightarrow G$ an open map?
Let $G$ be a Hausdorff topological group with center $Z$ and closed subgroup $H$. Suppose that $H.Z = G$. Is the product map
$$H \times Z \rightarrow G$$
necessarily an open map? That is, can we ...
- 6,180
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64
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On minimality of semitopological and quasitopological groups
The phenomemnon of minimality is well-studied in the realm of topological groups.
Let us recall that a topological group $X$ is minimal if each bijecive continuous homomorphism $h:X\to Y$ to a ...
- 42k
2
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66
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Two questions about subsequential spaces with countable $k$-network
This question was motivated by my answer to this MO question, which asked about the characterization of spaces that belong to the smallest class of topological spaces that is closed under taking ...
- 42k
2
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102
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Is this concrete set generically Haar-null?
This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete.
First we recall the definition of a generically Haar-null set in ...
- 42k
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81
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A semigroup property related to von Neumann regularity
A very common and useful notion in rings is that of von Neumann regular elements: those elements $a\in R$ such that there exists $b\in R$ satisfying $aba=a$. As this is a property defined solely ...
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50
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Nonautonomous wave equation of memory type
I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation
$$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$
This problem can be written ...
- 617
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169
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What is the difference between a monosemiring and a semigroup?
What is the difference between a monosemiring and a semigroup?
The following definitions are for clarity of my question.
A semigroup $S$ is a non empty set that satisfies closure and associativity ...
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63
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QF-3 monoid algebras
A finite dimensional algebra $A$ is called QF-3 in case the injective envelope of the regular module is projective. For example all Frobenius algebras are QF-3.
Given a monoid algebra $kG$ of a finite ...
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91
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Semigroups of nondecreasing functions
Consider some partially ordered set $(E,\leq)$. Assume either that it is countable with the discrete topology, or that it has some topology compatible with the order, preferably one that makes it into ...
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First Betti number of a Reeb graph is not greater than that of the space?
(I have asked this question at math stackexchange, it was upvoted but got no answers; maybe you can help.)
It is well-known that $\beta_1(R(f))\le\beta_1(X)$, where $\beta_1$ is the first Betti ...
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143
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How to unify these three classes of topological spaces?
For a cardinal $\kappa$ let $\Box^\kappa\mathbb R$ be the box-product of $\kappa$-many lines and $\boxdot^\kappa\mathbb R:=\{x\in\Box^\kappa:|\{\alpha\in\kappa:x(\alpha)\ne 0\}<\omega\}$ be the $\...
- 42k
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545
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On Kalai's $3^{d}$ conjecture
I just learned the existence of Gil Kalai's $3^{d}$ conjecture, which according to Wikipedia, is proven for $d$ at most $4$. It states that every $d$ dimensional polytope with central symmetry has at ...
- 6,984
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98
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If $H$ is an atomic, unit-cancellative monoid such that the set of atoms of $H$ is finite up to associates, then $H$ is BF
In a previous version of this post, $H$ was an atomic commutative monoid such that the quotient $H/H^\times$ is finitely generated, and I was asking if such conditions were enough for $H$ to be BF. ...
- 10.5k
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126
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Local cohomology with supports in a constructible set
Let $X$ be a topological space (I'm interested in the case of $X$ being a complex algebraic variety with the Zariski topology) and $Z$ a constructible subset (i.e. a finite union of locally closed ...
- 3,079
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65
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Splitting of ordinals of oscillation ranks of a Baire $1$ function
Denny and Tang proved that
Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$
Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and $\...
- 623
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35
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If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?
Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
- 3,079
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49
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Does each weakly feathered topological group admit an injective homomorphism into a feathered topological group?
A topological group $G$ is called
$\omega$-$\mathit{narrow}$ if for each non-empty open set $U\subset G$ there exists a countable subset $C\subset G$ such that $G=CU=UC$;
$\mathit{feathered}$ if $...
- 42k
2
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73
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Is there a star Lindelöf topological group which is not star countable?
I'm interested in this question:
Is there a star Lindelöf topological group which is not star countable?
A topological space $X$ is said to be star countable if whenever $\mathscr{U}$ is an open ...
- 621
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61
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Looking for a weakly Lindel\"of Tychonoff Moore non-ccc space
Is there a weakly Lindel\"of Tychonoff Moore non-ccc space?
Note that here ccc denotes the countable chain condition; a space $X$ is called weakly Linde\"of if for any open cover $\mathcal U$ of $X$ ...
- 621
2
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0
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217
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Constructible sets, I (Morphisms)
I have a number of questions on constructible sets. The first one is on morphisms: suppose $X$ and $Y$ are constructible sets, respectively in projective spaces $\mathbf{P}_1$ and $\mathbf{P}_2$ over ...
- 4,547
2
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62
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Extensions of an ideal-theoretic criterion for a monoid to be BF
Let $H$ be a multiplicatively written, commutative monoid. We denote by $H^\times$ the set of units (or invertible elements) of $H$, and by $\mathcal A(H)$ the set of atoms (or irreducible elements) ...
- 10.5k
2
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192
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Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
...
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