All Questions
5,185 questions
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Is the Upper Banach density always zero with respect to some sequence of Finite subset
The following question came to me while reading the paper 'Density in Arbitrary Semigroups' by Hindman and Strauss.
Question: Given an infinite subset $A$ of $\mathbb{N}$ such that $A^c$ is also ...
- 73
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298
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Fully faithful functor from schemes to spaces
Is there a fully faithful functor from the category of schemes
to the category of topological spaces and continuous maps (or some other sufficiently topological objects, e.g. smooth manifolds and ...
user137517
1
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151
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Clarification about the ϵ -net argument
I have been reading the paper Do GANs learn the distribution? Some theory and empirics.
In Corollary D.1, they reference the paper Generalization and Equilibrium in Generative Adversarial Nets which ...
- 19
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263
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Norm closure of $C_b^1(\mathbb{R})$
I want to determine what the closure of $C_b^1(\mathbb{R})$, the space of continuous differentiable functions with bounded derivative, with respect to the supremums norm is. I think that $\overline{...
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0
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29
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Localized connected expansions
Given a connected space, it is easy to tell if there is a connected expansion because maximal connected spaces (those admitting no finer connected topology) have the property that every dense subset ...
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0
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55
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Schemes for conditional distributions (monads)
(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.)
Suppose you have a monad ...
- 121
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111
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When a semigroup ideal is a determinantal ideal?
Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \...
- 113
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79
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Difference between planar sub-continua and sub-continua on the surface $\mathbb{T}^2$?
Can anyone tell me what is the essential difference between planar sub-continua and sub-continua of the torus? I will appreciate if you can give me some references.
- 97
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122
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Mackey topology characterising property
Let $V$ be a topological $k$-vector space.
Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all continuous linear functionals.
...
- 11
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90
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Cobordism of an annulus with a non-vanishing vector field
Let $M$ be a compact three-dimensional manifold with corners, which is a cobordism of the two-dimensional annulus. In particular, the codimension one boundary of $M$ consists of two copies of the ...
- 778
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127
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Explicit description of the scheme obtained by relative gluing data over a base scheme
I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...
- 453
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127
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How many two-dimensional space filling Hilbert-like curves are there?
I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...
- 355
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113
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Commutations of some limits and colimits in $\mathbf{CGWH}$
I know that finite limits do not commute with filtered colimits in general in $\mathbf{CGWH}$, nevertheless, do colimits commute with pullbacks, when we consider diagrams of the form $$\begin{matrix}&...
- 765
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630
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Is the boundary of an open set in a $\sigma$-space empty?
Recall that a Boolean space is a $\sigma$-space in case the closure of every open Borel set is open.
Let $\{B_i\}$ be a denumerable family of open-closed sets in a $\sigma$-space $X$. Then $\bigcup_i ...
- 175
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116
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Path connected without bounded path connected subset?
Question: Is there a path connected subset of $\mathbb R^2$, without any bounded path connected subset (aside from singletons)?
Motivation: If we replace "path connected" by "connected", then the ...
- 2,829
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126
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Determine all possible magnetic monopole of gauge theories
In Wikipedia, it states about the magnetic monopole of the gauge theory is determined by the fact:
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It ...
- 10.5k
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116
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A generalized Cauchy type functional equation
Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$.
Then is it true that $f(x+y)=f(x)...
- 1,209
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76
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Is there any characterization for lifting clopen subsets
Let $Y$ be a subset of a topological space $X$. We say that a clopen subset $L$ of $Y$ lifts to $X$ whenever there exists a clopen subset $H$ of $X$ such that $H\cap Y=L$.
Let $X$ be a compact and $...
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191
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Clopen subsets of a closed subspace of a spectral space
Let $X$ be a topological space. Set
$K(X) := \{ A\subseteq X\mid A$ is quasi-compact and open $\}.$ A topological space $X$ is called spectral,
if it satisfies all of the following conditions:
1) $...
- 11
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142
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Comparing Different Notions of Unicoherence in the Plane
Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...
- 439
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403
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Dual of $C(X)$ with the compact open topology
Let $X$ be a completely regular space and let $C_k(X)$ be the space of all continuous functions with the compact-open topology. If $X$ is completely metrizable, is the strong dual $C(X)^*$ the strong ...
- 19
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91
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Does There Exist a Planar, Linear, Triodic Tree-Like Continuum?
Motivated by Continuum image of line is chainable?
A planar continuum $X$ is a compact, connected subset of the plane. It is linear if there is a continuous bijection from $[0,1)$ onto $X$, for ...
- 439
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93
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An example of a Borel map of the first class
Let $X,Y$ be compact metric spaces, $2^X$ the set of all closed subsets of $X$ and $f:X\to Y$ be a 1st class Borel mapping.
Im trying to check Borel class of mapping $G:2^Y\to 2^X$. I submit it in a ...
- 21
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119
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A special topological property
Let us say a topological space $X$ is a countable union of second countable spaces if there exists a sequence of subsets $\{X_n\}$ of $X$ with $X=\cup X_n$ such that the relative topology on $X_n$'s ...
- 4,058
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280
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Comparing two $\sigma$-algebras
Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$.
Q. For which ...
- 4,058
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121
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A section over an orbit space
Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup.
Questions:
...
- 1,959
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50
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A criterion for metrizable topological spaces [duplicate]
Let $X$ be a topological space. True or false?
$X$ is metrizable if and only if it contains a sequence of metrizable spaces $\{X_n\}$ with $X=\bigcup X_n$!
- 4,058
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133
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When is a nested sequence of closed sets a colimit?
Let $X$ denote a topological space and $X_0\subset X_1\subset \ldots\subset X$ a nested sequence of closed subsets of $X$ such that $$ \bigcup_i X_i =X$$
It is easy to see that in the general case $X$...
- 1,174
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82
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What is known about the cohomology of the matrix monoid?
When I say the cohomology of a monoid, I mean that of its classifying space (considering the monoid as a category with a single object).
Let $M_n(R)$ be the monoid of matrices with matrix ...
- 1,726
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67
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Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$
Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
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75
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partially commutative like monoids [duplicate]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ (empty word) whenever $\{a,b\}...
- 1,269
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256
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partially commutative monoid [closed]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$...
- 1,269
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255
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Presentation of amalgamated sum as a quotient of the direct sum
I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf).
I'm trying to understand why the amalgamated sum of ...
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113
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Continuous functions "sharing" a point
Let $(X,\tau)$ be a topological space. By $\text{End}(X)$ we denote the collection of all continuous functions $f:X\to X$. We say $f,g\in \text{End}(X)$ share a point if there is $x\in X$ such that $f(...
- 48.9k
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109
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Toral subgroup acting regularly on the homogeneous space
Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $...
- 1,959
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112
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Question regarding the image of a polynomial map containing a small box
I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously.
Let $\delta, \varepsilon > 0$.
Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
- 3,625
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112
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Is the topology generated by the complements of analytic subsets strictly coarser than the Euclidean topology in dimensions $\geq 2$?
Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$ and let $N\geq 2$. Similarly to the construction of the Zariski topology, take the collection of zero sets of $\mathbb{K}$-analytic functions to ...
- 7,127
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A exemple of a strongly-continuous contraction semigroup : how to prove the contraction?
I am trying to prove that $P_t := e^{\lambda t (P-I)}$ (where $Pf:= \int f(y) P(\cdot , dy)\in \mathcal{C}_0(\mathbb{R}^d)$, for $f\in \mathcal{C}_0(\mathbb{R}^d)$, $P$ being a probability kernel), is ...
- 111
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207
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free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$
I want to construct free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$.
For $n=2m-1$, consider $S^n ⊂ C^m$. Then $S^1$ freely act on $S^n$ by $(ξ, (z_1 , z _2 , · · · , z _m )) → (ξz_1 , ξz_2 ,...
- 395
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29
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Generating larger atoms from smaller ones in a simple $\text{C}_0$-monoid
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (...
- 10.5k
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125
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Characterization of topological spaces
Apparently, the following is an equivalent characterization of topological spaces: consider a relation $\ll$ on the power set $\mathcal{P}(X)$ of a set $X$, with the following axioms:
1) $\emptyset\...
- 163
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128
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The group of polynomial homeomorphism of the plane
Let $G$ be the set of all homeomorphisms $f$ of $\mathbb{R}^2$ such that
both $f$ and $f^{-1}$ are polynomial maps.
We equip $G$ with the compact open topology and the obvious group ...
- 366
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47
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Is the minimality of complete topological groups recognizable by closed separable subgroups?
A topological group is called minimal if it admits no strictly weaker Hausdorff group topology.
By Prodanov-Stoyanov Theorem, a complete Abelian topological group is minimal if and only if it is ...
- 42k
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206
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A reasonable topology on the automorphism group of an $\omega$-narrow topological group?
For a topological group $X$ by $Aut(X)$ denote the group of topological isomorphisms $h:X\to X$. If $X$ is compact then the compact-open topology turns $Aut(X)$ into an $\omega$-narrow topological ...
- 42k
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56
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A topology for which symplectic forms are dense in skew forms
Let $V$ be a vector space over an algebraically closed field. Let $S$ denote the vector space of skew-symmetric bilinear forms on $V$. When $V$ is finite dimensional the subset of $S$ consisting of ...
- 599
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105
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The inverse image of a Noetherian topological space
A topological space $X$ is called Noetherian if
closed subsets satisfy the descending chain condition, equivalently,
the open subsets satisfy the ascending chain
condition.
Let $A$ and $B$ be ...
- 11
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82
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Topology of the algebra $\mathbb{C}\{A\}$ for a LCA group $A$
Let $\mathcal{A}$ be a complex associative Hausdorff topological algebra, and let $A\subset\mathcal{A}$ be a locally compact Abelian (LCA) subgroup (multiplicative). The linear span $\mathbb{C}\{A\}$ ...
- 1,959
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149
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Groups and non-trivial finite topologies
Recently, I have been "updating" myself in the field of topological groups, and, in doing this, I remembered some questions I had a few years ago that I never solved.
First, is there any application ...
- 3,107
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50
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Closedness of the range of the distorsion of the multiplicative monoid of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is an element $x \in H \setminus H^\times$ such that $a \ne xy$ for all $x, y \in H \setminus H^\times$, where $H^\...
- 10.5k
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162
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Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
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