All Questions
5,184 questions
4
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Is each typical map on the $n$-cube strongly rigid?
This question is inspired by this (still unanswered) MO-post.
A function $f:X\to Y$ between topological spaces is called strongly rigid if every continuous self-map $h:X\to X$ with $f\circ h=f$ is the ...
- 42k
1
vote
0
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181
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Second homology group of a presentation complex
I am trying to learn results related to the presentation complex of a group and I am new to this subject. So I apologize if the questions are silly.
Given a finite group $G$, and a presentation $P$ of ...
- 179
2
votes
1
answer
264
views
Formula for the Euler characteristic of a local system on $\mathbb{P}^1$
Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.
Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...
- 149k
1
vote
1
answer
204
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Name of a space with both a topology and a metric that are not compatible?
Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$.
Is there a canonical name for such a structure (maybe ...
- 790
-1
votes
2
answers
260
views
Function space and contractibility
$\DeclareMathOperator\map{map}$I have the following question:
Let $X$ and $Y$ be topological spaces. Let $\map(X,Y)$ denote the space of non-constant continuous functions from $X$ to $Y$. Suppose ...
- 3,948
2
votes
0
answers
161
views
Embedding a monoid into a group via its monoid ring
Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...
CommunityBot
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4
votes
0
answers
136
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Is there a condensation of a closed subset of $\kappa^\omega$ onto $\kappa^\omega\setminus A$ …?
Let $\aleph_1\le\kappa<c$ and $A\subset \kappa^{\omega}$ such that $\lvert A\rvert\le\kappa$.
Is there a condensation (i.e. a bijective continuous mapping) of a closed subset of $\kappa^\omega$ ...
- 1,101
13
votes
5
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1k
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A generalization of metric spaces
Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i....
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10
votes
1
answer
569
views
Quotient of complete topological ring
Let $G$ be a complete topological group (meaning that every Cauchy net has a unique limit), and $H\unlhd G$ a closed normal subgroup. If $G$ is first countable (equivalently, metrizable), then $G/H$ ...
- 16.5k
7
votes
2
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587
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Representation theory of the full linear monoid
The full linear monoid $M_N(k)$ of a field $k$ is the set of $N \times N$ matrices with entries in $k$, made into a monoid with matrix multiplication.
A representation of $M_N(k)$ on a vector space $V$...
- 11.2k
4
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0
answers
74
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Is each TS-topologizable group TG-topologizable?
Definition 1. A topology $\tau$ on a group $X$ is called
$\bullet$ a semigroup topology if the multiplication $X\times X\to X$, $(x,y)\mapsto xy$, is continuous in the topology $\tau$;
$\bullet$ a ...
- 42k
2
votes
1
answer
853
views
The Borel sigma-algebra of a product of two topological spaces
The following problem arose while trying to justify some "known results" in abstract harmonic analysis on noncommutative groups, for which I couldn't find explicit statements in the ...
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4
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0
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170
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Corollaries of Kleene's Theorem (Regular Languages)
Kleene's theorem that finite automata (specifically, nondeterministic) are expressively equivalent to regular expressions seems to be a powerful and not immediately obvious tool for untangling the ...
- 429
4
votes
0
answers
157
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On skew monoid rings and skew ordered series rings
To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...
- 10.5k
2
votes
1
answer
183
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On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$
Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$.
For which $X$'s is it true that $J_A+J_B=J_{A\cap ...
- 11.4k
3
votes
2
answers
472
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Regularity of lipschitz and derivable function
Let be lipschitz $f$ on $[0,1]$ and everywhere derivable. Is it true that $f\in C^1([0,1])$ ?
- 3,312
17
votes
3
answers
2k
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Decidability of tiling R^2
Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane?
I know the ...
- 39.9k
8
votes
1
answer
355
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State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"
The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye :
"If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...
- 18.6k
2
votes
2
answers
519
views
How to use that the Hessian is negative definite in this proof
Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...
- 442
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votes
1
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243
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Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ has empty interior [closed]
Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? ...
CommunityBot
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0
answers
52
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Can we decompose an increasing net of functions into two increasing nets with prescribed supports?
Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...
- 5,529
2
votes
0
answers
66
views
Separating property of a finite union of topological disks
Let $X$ be a topological $2$-sphere. Let $D_1, D_2, \dots, D_n \subset X$ be a finite family of closed topological disks (i.e. sets homeomorphic to the closed unit disk). Let $\mathcal{U} = \bigcup_{1 ...
- 111
4
votes
1
answer
111
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A neighborhood of a 2-disc $D\subset\Bbb R^4$ that tapers off towards the boundary?
I am given a PL 2-disc $D\subset\Bbb R^4$ (everything PL from here on) and I need a "neighborhood" $N\simeq B^4$ (PL-homeomorphic to a 4-ball) so that $\partial N\cap D=\partial D$.
If I got ...
- 10.5k
-4
votes
2
answers
405
views
Do these irrationals exist?
An irrational $a$ verifies : $\{a\times n+k;(n,k)\in\mathbb Z^2 \}$ is dense in $\mathbb R$.
If you take $a$ universe then : $\forall b\in \mathbb N^*, \{a\times n^{b}+k;(n,k)\in\mathbb Z^2\}=A(a,b)$ ...
- 2,621
4
votes
1
answer
1k
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Condensed mathematics
I have a little technical question on Peter Scholze's lectures on condensed mathematics.
On page 12, right above the Proof of Theorem 2.2, he says that for extremally disconnected sets the condition (...
- 63.9k
2
votes
1
answer
443
views
Possible characterisation of compactly generated weakly Hausdorff spaces
Is it true that, in the category $\mathbf{Top}$ of topological spaces and continuous maps, the compactly generated weakly Hausdorff spaces are precisely the spaces arising as filtered colimits of ...
- 35.5k
25
votes
2
answers
1k
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The number of polynomials on a finite group
A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
- 42k
6
votes
0
answers
190
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The highest degree of a polynomial on a finite group
This question is motivated by the comments and the answer to this MO-question.
First let us recall some definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\...
- 42k
0
votes
0
answers
41
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Polyextremal groups
A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form
$f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
- 42k
1
vote
0
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99
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Density of Lipschitz functions in Bochner space with bounded support
Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...
- 63.9k
3
votes
1
answer
132
views
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?
Let $F\subset \Bbb{R}$ intersect every closed uncountable subsets of $\Bbb{R}$.
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of ...
- 42k
8
votes
1
answer
234
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If $M$ is contractible manifold and $X\subset \partial M$, does the cone over $X$ embed in $M$?
Let $M$ be a compact contractible manifold, $X\subset\partial M$ and $C_X$ the cone over $X$.
Question: Is it true that $C_X$ embeds in $M$ with its boundary $\partial C_X$ mapped to $X\subset \...
- 13.6k
2
votes
1
answer
507
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(Dis)prove : if every function with closed graph are continuous then the target space is compact
$(X, \tau_X) $ and $(Y, \tau_Y) $ be two topological spaces.
$\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $.
Question : Does this implies $(Y, \tau_Y) $ is compact?
...
- 2,856
5
votes
3
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598
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Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex?
Does a compact semilocally simply connected geodesic space have the homotopy type of a compact CW complex? Actually what I'd like to know is whether the fundamental group of such a space is finitely ...
- 1,726
6
votes
1
answer
191
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Steinhaus number of a group
$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$.
Let $\mathcal A_X$ be the family of ...
- 42k
1
vote
0
answers
53
views
The "hyperbolicity preserving" probabilities
A classical fact (due to Polya ?) is that if $P\in{\mathbb R}[X]$ has only real roots (one says that $P$ is hyperbolic), and $a$ is a real number, then the roots of
$$L_aP(X):=\frac12(P(X+ia) +P(X-ia))...
- 52.3k
2
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0
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103
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$n$-connected spaces (terminology)
A graph is called $n$-connected if it remains connected after removal $\le n$ vertices.
Question. What is the name of an analogous property of topological spaces: a space that remains connected after ...
- 42k
0
votes
2
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503
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A Jordan arc in the unit disk
Let $D$ be the open unit disk, and $J$ a Jordan arc (that is, a homeomorphic copy of $[0, 1]$) that lies in $D$, except $J(0)$ lies on the boundary of $D$, say $J(0)=1$. I would like to see that $D\...
- 63.9k
2
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0
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181
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So many types of subwords! How are they called?
Let $\mathscr F(X)$ be the free monoid on an alphabet $X$, the carrier set of $\mathscr F(X)$ being the union of $X^{\times k}$ (the Cartesian product of $k$ copies of $X$) as $k$ ranges over $\mathbb ...
- 10.5k
7
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2
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537
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How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$?
I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere.
Here $\Omega$ is any ...
- 7,817
2
votes
0
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134
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Subsets of $\mathbb{S}^n$ fixed by an orientation-reversing self-homeomorphism — Part 1
Call a subset $Z$ of $\mathbb{S}^n$ ambiently-reversible, if there is an orientation-reversing self-homeomorphism $h: \mathbb{S}^n \to \mathbb{S}^n$ fixing $Z$ pointwise.
Question 1: Which subsets of ...
- 63.9k
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Subsets of $\mathbb{S}^n$ fixed by an orientation-reversing self-homeomorphism — Part 2
This is another special case of this question.
Recall that we call a subset $Z$ of $\mathbb{S}^n$ ambiently-reversible, if there is an orientation-reversing homeomorphism $h: \mathbb{S}^n \to \mathbb{...
- 63.9k
19
votes
2
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1k
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Are there space filling curves for the Hilbert cube?
There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes.
So my question is: ...
- 4,713
3
votes
0
answers
106
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Two topologies on the space of maps from an algebraically closed field to a projective variety
This question is related to this one but I have written this in a self-contained manner.
All varieties are complex varieties.
For quasi-projective variety $U$ and a projective variety $X$ we can ...
- 5,901
6
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0
answers
189
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What is a non-smooth connection?
Let $p : E \to B$ be a map of topological spaces, and $p^I : E^I \to B^I$ the induced map of path spaces. Let $Cocyl(p) = B^I \times_B E$ be the space of paths $\beta$ in $B$ equipped with a lift of $\...
- 64k
0
votes
0
answers
62
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To find a DFT for complex functions on a semigroup
For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
13
votes
3
answers
619
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Extending homeomorphisms from closed countable sets to S^2
Let $A, B \subset S^2$ be closed, countable sets and $\phi \colon A \rightarrow B$ be a homeomorphism. Can we extend $\phi$ to a homeomorphism from $S^2$ to itself?
It is well-known that the answer ...
- 1,936
10
votes
1
answer
183
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Do any two closed, countable, homeomorphic subsets of $\mathbb{R}^3$ have homeomorphic complements?
Let $A,B$ be two closed, countable, topological subspaces of $\mathbb{R}^3$ homeomorphic to each other. Must their complements $\mathbb{R}^3 - A, \mathbb{R}^3 - B$ be homeomorphic to each other?
(The ...
- 1,936
3
votes
0
answers
88
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Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?
Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...
- 13.6k
0
votes
1
answer
291
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Tensor product is complete?
Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...
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