All Questions
5,184 questions
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139
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Computationally intractable orbit of a monoid action on a finite set
Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$.
A characterization of $M_n$ is an algorithm that takes an integer $...
- 1
0
votes
1
answer
254
views
Continuity of Kakutani fixed points
Let $X$ be a compact and convex space and let $T=[0,1]$ be some parameter space. Let $F:X\times T\rightrightarrows X$ be a correspondence that is compact-valued, convex, and upper-hemicontinous. By ...
- 229
0
votes
1
answer
969
views
Is the pointwise supremum of a continuous function continuous?
Suppose $f(x , y)$ is continuous in both variables. For any $\epsilon > 0$ and some $y_0$, let $h_{\epsilon}(x) = \max_{y^{'}: \| y^{'} - y_0 \| \leq \epsilon} f(x , y^{'})$. It seems to me that $...
- 3
0
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1
answer
225
views
Convergence in compact-open topology on the Sierpiński space
Question:
Equip $\{0,1\}$ with the Sierpiński topology $\{\{1\},\{0,1\},\emptyset\}$, let $X$ be a compact metric space, and equip $C(X,\{0,1\})$ with the compact-open topology. Let $\{B_n\}_{n=1}^{\...
- 5,405
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1
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86
views
Lattice of functions and their minimal separating set upto topological equivalence
There is a very wide series of questions I have been thinking about and I am wondering if there is any literature on this type of structures.
Let's start with the set of all functions $F: \mathbb{R} \...
- 39
0
votes
1
answer
74
views
What is the source to find cardinal invariants for a function space C(X, Y), equipped with uniform or fine topology?
I would like to know about the technique to check the cardinality properties for the function space C(X, Y), where X is a tychonoff space and Y a metric space, equipped with uniform or fine topology.
0
votes
1
answer
81
views
Ultrabornological representation for the space of uniformly continuous functions?
Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space
$$
C_{\omega_i}(\mathbb{R}^n,\...
- 5,405
0
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1
answer
200
views
The rank of a semigroup
Let $S$ be a finite noncommutative semigroup(without identity) with a subset $M$ such that $\langle M \rangle =S$. If every element of $M$ is indecomposable in $M$, i.e. for any $a \in M$, there are ...
0
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1
answer
136
views
A link between continuity and 0-borelian? [closed]
Is it true that :
1/ if $f$ real continuous and $O$ an open set then $f(O)$ is a 0-borelian?
2/ if $A$ a 0-borelian set then there exists $f$ real continuous and $O$ an open set with $A=f(O)$?
$B$ ...
- 4,074
0
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1
answer
111
views
If a topological space (X,T) is completely normal, and if we double the point of X, is the resulting space also completely normal? [closed]
I have a question on my proof of the following lemma, and I'd like to know if my answer is correct.
Lemma. Suppose $(X,T)$ is any completely normal topological space. Let's double the points of $X$, ...
- 73
0
votes
1
answer
91
views
Topologically transitive dynamical system mapping space into ball
Let $X$ be a separable Hausdorff topological space and $\phi \in C(X,X)$ be a topologically transitive map. Further, let $V$ be a fixed non-empty open subset of $X$. Then does there necessarily ...
- 5,405
0
votes
1
answer
223
views
Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?
Let $X$ be a separable Banach space and $D\subseteq X$ be a
proper, connected, and dense $G_{\delta}$ subset of $X$,
$X-D$ is $\sigma$-porous.
Then is $X-D$ contained in a finite-dimensional ...
- 5,405
0
votes
1
answer
296
views
Multiplicative monoid of ring modulo units
Let $M = \mathbb{Z}[\phi] \setminus \{0\}$ be the multiplicative monoid of the ring $\mathbb{Z}[\phi]$ with $\phi = \frac{1+\sqrt{5}}{2}$ the golden ratio.
We define the equivalence relationship $x\...
- 943
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1
answer
134
views
Lower semicontinuity of a multi-valued map $F:X\to 2^Y$ in term of net
Let $X,Y$ be two Hausdorff spaces and $F:X\to 2^Y$ be a multi-valued mapping. We says that $F$ is lower semicontinuous at $x_0\in X$ if for each $y_0\in F(x_0)$ and any neighborhood $U\in \mathcal N(...
- 1,245
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votes
1
answer
183
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Sufficient and necessary condition for the global uniqueness of fix-points
https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf
This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if ...
- 263
0
votes
1
answer
60
views
Empty interior lack of minima
Suppose that $U \subseteq \mathbb{R}^d$, and satsifies
$U$ is dense in $\mathbb{R}^d$,
U has empty interior,
Then is it possible that
$$
\inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x),
$$
...
- 5,405
0
votes
1
answer
75
views
Existence of certain subsemigroups of $C(K, K)$ for compact Hausdorff spaces $K$
Let $K$ be a compact Hausdorff space. I'm wondering: Does there always exist a subset $J \subseteq C(K, K)$ such that:
$J$ is closed under composition,
there is an element $f \in C(K)$ such that the ...
- 741
0
votes
1
answer
88
views
Finite pre-images implies (local) branch cover?
Let $M_{1},M_{2}$ be (possibly non-compact) 2-dimensional, connected, smooth, orientable manifolds of finite topological type. Suppose you have smooth, surjective map $F:M_{1} \rightarrow M_{2}$, and ...
- 6,995
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1
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185
views
Borsuk–Ulam theorem on the sphere with expluded poles [closed]
Consider a sphere without two poles $U^2$. Will Borsuk–Ulam theorem still work, i.e. $\forall$ continuous functions $f:U^2 \rightarrow \mathbb{R}^2 ~\exists x \in U^2$ such as $f(-x)=f(x)$?
- 415
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1
answer
163
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Is there a Tychonoff space $X$ such that ....?
$X$ is not a separable submetrizable, i.e.($iw(X)>\omega$) $X$ has not a countable injective weight.
There is a Baire isomorphism 1-class between $X$ and a separable metrizable space $Y$.
- 1,101
0
votes
1
answer
129
views
Fundamental Surfaces in 3-manifolds
Given a 3-manifold $M$ with a triangulation $T$, will every essential surface in $M$ be a fundamental one? If not, then what are the conditions on $T$ so that these essential surfaces become ...
- 1
0
votes
1
answer
130
views
Bounded components of the complement [closed]
Hi everyone: Suppose $A\subset \mathbb{R}^{m}$ ($m>1$) is a closed set with empty interior. Which are the necessary and sufficient conditions that $A$ have a neighborhood $V$ such that every ...
- 411
0
votes
1
answer
63
views
Monoid morphisms satisfying a decomposition condition
Let $A$ and $B$ be monoids, let $f\colon A\to B$ be a morphism of monoids. The following pair of conditions emerged naturally in my research:
For all $a\in A$ and $b_1,b_2\in B$ such that $f(a)=b_1....
- 327
0
votes
1
answer
183
views
Partially ordered set of compatible topologies
Let $X\neq \emptyset$ be a set and let ${\cal J} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say that a topology $\tau$ on $X$ is ${\cal J}$-compatible ...
- 48.9k
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1
answer
83
views
Congruences of abelian monoids which can be extended to (ideal) congruences of polynomials
Some weeks ago I asked the same question at [math.stackexchange][1] but I have not gotten any feedback. The flavour of the question (but see the details later) is about whether to understand ...
- 173
0
votes
1
answer
119
views
Rank of a generall linear group over a finite field [closed]
What is the rank (minimal number of group generators) of the group $GL(n,F)$, when $F$ is a finite field of odd order? I found that $SL(n,F)$ is $2$, but I can't find this information.
- 19
0
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1
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52
views
Compact $R_1$-spaces
A space $(X,\tau)$ is said to be $R_1$ if for all $x,y\in X$ with $cl(\{x\}) \neq cl(\{y\})$, there are disjoint open sents separating $cl(\{x\})$ and $cl(\{y\})$.
If $X$ is compact and $R_1$, does ...
user62017
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1
answer
57
views
If $X$ has the "discrete" covering property, how about $X^2$?
We say that a space $X$ has covering property (C) if the following holds:
(C) For any open cover ${\cal U}$ of $X$ there is a closed discrete set $D\subseteq X$ and a map $\varphi: D\to {\cal U}$ ...
user70876
0
votes
1
answer
299
views
Can the image of a disk have nontrivial Hausdorff measure for $1 < d < 2$?
I am reading a blog which talks about a $C^1$ diffeomorphism $f: \mathbb{D}\{ x^2 + y^2 < 1\} \to \mathbb{R}^2$ and estimates the Hausdorff dimension of its image $\mathcal{H}_\sqrt{2}^d (f(\mathbb{...
- 22.8k
0
votes
1
answer
114
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Priestley topologizability and connected components
This question is in the spirit of another older question.
We say that a poset $(P,\leq)$ is Priestley-topologizable if there is a topology $\tau$ on $P$ such that $(P,\leq,\tau)$ is a Priestley space....
user70876
0
votes
1
answer
163
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Existence of half-planes with respect to regular open sets of the Euclidean plane
I initially asked this question at math.stackexchange.com but there was no reaction, so I thought this may be a good idea to transfer it to mathoverflow.net
Let $\langle\mathrm{r}\mathscr{O},\mathord{...
- 1,112
0
votes
1
answer
172
views
Can we build a continuous function from "fibers"/preimages defined over a topological base?
I am looking for some proof insight or literature references for a statement which, if it's actually true, is probably a pretty trivial thing. I hope the question has not been asked here before, and ...
- 273
0
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1
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452
views
Relative interior and dense subsets
(This is a cross-post from here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ by $$\text{ri}(S)=\{s\...
- 215
0
votes
2
answers
212
views
Is there a normal space that is not uniformly normal
Let $(X,\mathcal D)$ be a uniform space and $A,B\subseteq X$. Let's say $A$ is uniformly inside $B$ and write $A\le B$ iff there's some entourage $D$ for which
$$(\forall a\in A)(D[a]\subseteq B)$$
A ...
user31967
0
votes
1
answer
252
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Which algebra of functions can be represented as $C(X)$
I don't know if this problem is known or not, so any information would be appreciated:
Let $\cal A$ be an $\Bbb{R}$-algebra of (not necessary continuous) real valued functions defined on a ...
- 9
0
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1
answer
151
views
Is there any result concerning on the metric dimension of inverse limit?
To be specific, my question is as follows:
Question: Let $X$ be an inverse limit of compact metric spaces $(X_i, d_i)$, then does it hold
$\dim(X, d) \leq \sup_i \{\dim (X_i, d_i)\}$ for some ...
0
votes
1
answer
402
views
A question on cofinite topology.
Let $X$ be a countably infinite (or larger) set with the cofinite topology. for every $x\in X$ is there exists a family $\xi\subset\tau$ such that $\lbrace x\rbrace=\bigcap\xi
$ ? If the answer is yes,...
- 13
0
votes
1
answer
305
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Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
- 10.5k
0
votes
1
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94
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What is the way to after finding Cohen-Macaulay semigroup as ring of monomial?
http://people.missouristate.edu/lesreid/reu/2007/PPT/robin.ppt
it said missing part inside and not missing part outside is non Cohen-Macaulay semigroup
which determine whether is missing in the most ...
- 1
0
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1
answer
231
views
A question on linearly lindelof space
Let $X$ is a linearly lindelof subspace of $Z$ and $b$ is not $\omega$-separated from $X$, i.e., for any closed $G_\delta$ set $P$ of $Z$ which contains $b$, $P\cap X \not=\emptyset$. If $\tau < \...
- 654
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1
answer
851
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Example of a completely regular spaces
A topological space $X$ is an $EF$-space if if for any
two collections $\mathcal{U}$ and $\mathcal{V}$ of clopen subsets
of $X$ with $\bigcup \mathcal{U}\cap \bigcup
\mathcal{V}=\emptyset$, we have $\...
- 192
0
votes
1
answer
87
views
Question regarding closure of sets defined by the vanishing of holomorphic functions
Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, g$...
- 3,245
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1
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341
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Length of intersection of intervals
Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...
- 500
0
votes
3
answers
404
views
Some Questions about zero-dimensional subsets of the unit interval related to cantor set
Let $\mathbb{P}$ denote the set of all irrational numbers in the open segment$(0 , 1)$. let $K$ be the intersection of $\mathbb{P}$ and the standard cantor set and $H=\mathbb{P}-K$. as you know these ...
- 1,788
0
votes
1
answer
493
views
Sheaf of sections and local triviality
This is probably not a research level question, I'm sorry if it is inappropriate. I'm reasking here this question on math.se.
Suppose that $\xi: E \to B$ is a bundle (by which I mean simply a ...
0
votes
1
answer
2k
views
What does it mean to have Zero Density (mathimatically) [closed]
I read a question that asked "prove that the set of all positive integers expressible as the sum of two integers square has zero density." Now I was under the impression that something was dense iff ...
- 93
0
votes
1
answer
280
views
"Skein" equations sets that can reduce any graph
Consider for example the approach to the Kuperberg G2 or the Yamada polynomial. I don't know whether relations of graph (in contrast to knot) theory are also usually called "skein" but
I simply carry ...
- 4,829
0
votes
2
answers
796
views
Extending Continuous Sublinear maps on dense subsets of a Banach space
Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying:
(1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||.
Please prove ...
- 11
0
votes
2
answers
643
views
Collinear vertices and definition of k-simplex
On page 120 of his Basic Topology, Armstrong defines the $k$-simplex in $\mathbb{E}^n$ with verices $v_0,\ldots,v_k$ to be complex hull of said vertices. (A similar definition is given on Wikipedia).
...
- 3,079
0
votes
1
answer
147
views
Small set of acts over a countable monoid?
Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?
- 11