All Questions
1,339 questions with no upvoted or accepted answers
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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 ...
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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 ...
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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 ...
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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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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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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(...
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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 ...
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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 ...
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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 ,...
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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 $H$ (...
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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\...
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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 ...
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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 ...
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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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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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80
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Extending maps to disc homeomorphisms isotopic to the identity
Consider the closed unit disc $\mathbb D^n$ in $\mathbb R^n$ and its closed subdisc $D$ centered at the origin with radius $1/2$. Denote by $V$ the interior of $\mathbb D^n$. I wonder whether the ...
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58
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A class of finitely generated semigroups
Let $G$ be a finitely generated group with a probability measure $\nu$. Suppose we have a finite first moment function $f:G\to \mathbb{R}$, i.e, such that $\Sigma_{g\in G}f(g)\nu(g)< \infty$. Then, ...
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152
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Stone cech compactification of a zero dimensional topological space
Let $X $ be a zero dimensional topological space, that is, a topological space with a basis of clopen sets. Is there any characterization for the ston cech compactification for such a space?
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107
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Which kind of functions satisfy this property?
We look to the Banach space $L^{\infty}([0,1])$ with the well-known norm on it and the weak-*-topology (which is in fact locally convex), hence $f_n\rightarrow f$ in the weak *-topology iff $\int\...
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139
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Is this a known compactification of graphs?
Let $G$ be a locally finite, connected, and infinite graph. Let $\Omega(G)$ its set of ends. Let $|G|$ be the Freudenthal compactification of $G$. Let $P_{\Omega}$ be a partition of $\Omega(G)$. ...
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81
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A consecutive resolution of continum algebras to a simple continum algebra
Motivated by classical Gelfand Naimark duality, the correspondence between the category of commutative $C^{*}$ algebras and the category of locally compact Hausdorff spaces, we ...
- 356
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220
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About the projection on the unit sphere
Let $H$ be a Hilbert Space and let $A\subset H$ be a connected set such that any two elements of $A$ are linearly independent and also $A^{\bot}=\left\{0\right\}$ (this seems to be immaterial). Is ...
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251
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Copylefted introduction to topology
Is there a textbook in topology with a copyleft license?
$$ $$
- 45k
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138
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Finding a metric on a topological space with prescribed isometry group
Let $X$ be a (sufficiently nice) topological space and let $\mathcal{F}$ be a group of homeomorphisms of $X$. Assume that $\mathcal{F}$ is also closed under point-wise convergence. I would like to ...
- 1,159
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77
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Random variables with values in binary operations or in topologies of a certain set $X$
I wonder if the following situations have already been considered by mathematicians :
Random variables with values in a set of binary operations endowed
with a certain topology (or just with a $\...
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639
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What is the real name of this relation and operation on a particular set of maps between cancellative monoids?
Let $A,B$ be cancellative monoids and define a transducer as a map $f\colon A \rightarrow B$ such that $f(1)=1$ and for all $a_1 ,a_2 \in A$, there exists a $b \in B$ such that $f(a_1 a_2)=f(a_1) b$. ...
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133
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Standard proof that cyclic ordering of edges is preserved under planar graph homotopy?
I have several questions about the following theorem statement:
Thm: Let $G = (V, E)$ be a planar graph, and let $\varphi_0 : G \rightarrow \mathbb{R}^2$,
$\varphi_1 : G \rightarrow \mathbb{R}^2$ be ...
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81
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Homotopy invariant deletions of open faces of simplicial complexes
Given a finite simplicial complex (as a topological space) $\Delta$ and a face $\tau$, suppose we delete the interior of $\tau$ (a point if $\tau$ is a vertex, otherwise homeomorphic to an open ball ...
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Inverse limits of the interval with a single bonding map below the identity
My question is as follows.
QUESTION. Is there a topological description of the class of arc-like continua that arise as inverse limits of $[0,1]$ with a single continuous surjective bonding map $f\...
- 6,548
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182
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$\mathbb E$-descent maps in topological spaces in terms of different sites?
The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories.
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90
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The role of absolute continuity in stochastic ordering defined over sets of probability distributions
This question is about a claim given in this paper (page 261, the remark), but without any proof.
It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ (...
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305
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Alternative representation of $C_c(X)$ as inductive limit
CORRECTION: As Simon Henry points out in the comments, there is a problem in the construction: the maps $\varphi_n$ are not necessarily linear.
Under some additional constraints on the space (e.g. $X$ ...
- 1,773
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152
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Does bounded and closed equal compact for sets of Borel probability measures?
Equip the space of Borel probability measures on a fixed closed subset X of the s-dimensional Euclidean space with the topology induced by weak convergence of probability measures. In this setting, ...
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130
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Not normal connected component of a right topological group
Let $\cal T$ be a locally compact topology on a group $G$ and $(x,y)\mapsto xy^{-1}$ be continuous at $(1,1)$ and for every $a\in G$, $x\mapsto xa$ be continuous everywhere with respect to $\cal T$.
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- 1,145
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82
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Name for a type of weak path connectedness?
The following topological property arose in the context of my friend Don Hadwin's operator theory research, and he asked me to ask here if the property occurs in the literature and has a name.
...
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96
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Is there another equivalence relation on based maps between spheres which form the same graded ring as the homotopy groups?
Let $\sim$ be an equivalence relation on continuous based maps from $S^k$ to $S^n$, where $k$ and $n$ range over the positive integers.
Suppose that
Given maps $f, f^\prime: S^k \to S^n$ and $g, g^\...
user2529
1
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261
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The closure of a set of closed points
Let $X $ be a compact non-Hausdorff topological space with the following property: for every infinite subset of closed points, say $\{x_i\}_{i \in I}$, there exists $j\in I$ such that $x_j$ is in the ...
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293
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Examples of value quantales
In his paper "Quantales and continuity spaces" R. C. Flagg gives the following examples of value quantales: the lattice $\bf{2}$ of truth values with usual addition, the lattice $\mathbb{R}_{+}$ of ...
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127
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Category-theoretic characterization of zero-dimensional spaces
Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
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