All Questions
5,184 questions
3
votes
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307
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product spaces of rationals
Let $Q$ follow subspace topology from $R$
Then I think it is true that $Q^n$ and $Q^m$ (with product topology) are not homeomorphic.I also think it will be possible to define "rational" homotopy ...
- 153
6
votes
1
answer
517
views
Growth zeta-functions of regular languages
Dear All,
my following question may be known and ought to be known, so in case it is folklore please could you give me the references.
To start, it is obvious that growth of rational languages are ...
- 1,437
3
votes
3
answers
446
views
A question concerning the isomorphic type of continuous functions
let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside a bounded open interval containing zero (depended on $f$). Is it possible to consider $S$ as (...
user39207
5
votes
2
answers
537
views
If $k[S]$ is noetherian, is S finitely generated?
Let $S$ be a semigroup. If $S$ is abelian, then it follows that the semigroup algebra $k[S]$ is finitely generated if and only if $S$ is.
What if we relax the condition on $k[S]$, so that $k[S]$ is ...
- 11.6k
-11
votes
1
answer
2k
views
Union of uniformly connected sets
I will call a set uniformly connected regarding some uniform space when it is connected regarding every entourage of this uniform space (entourages are considered as digraphs and it is taken strong ...
- 765
2
votes
1
answer
182
views
Terminology for the equation $a=a+b$ in commutative semigroups
Let $(S,+)$ be a commutative semigroup. For $a,b\in S$ consider the equation $a=a+b$. Does such a relation between the given $a$ and $b$ have a name? I am currently using such equations quite often ...
2
votes
2
answers
657
views
Varieties, Frechet Completions, and Regular Functions
Take an algebraic variety $V$, and its set of smooth functions $C^{\infty}(V)$. One can endow $C^{\infty}(V)$ with a canonical locally convex topology (the seminorms are defined using the local ...
- 1,462
0
votes
1
answer
425
views
Property of Mrowka
A topological space $X$ satisfies Property $K_1$ (Property of Mrowka) if the closure of the union of arbitrarily many $G_\delta$ sets of $X$ coincides with its sequential closure (the sequential ...
- 35
-5
votes
1
answer
483
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For every proximity, does there exist a uniformity which generates this proximity?
For every proximity, does there exist a uniformity which generates this proximity?
This question may be generalized for different generalizations of proximities and uniformities. In fact I need it ...
- 765
2
votes
1
answer
136
views
A souped-up version of a question asked previously about uncountable subsets of topological spaces
Let T be an uncountable Hausdorff space. The following property of T will be referred to as "property P". If S is any uncountable subset of T, then the set of all points of S that are not limit points ...
- 6,215
1
vote
1
answer
582
views
Lifting identities of formal power series
I am looking for a possibly general class of algebraic structures (maybe special topological rings) in which one can deduce identities of concrete power series from formal ones. This class should ...
- 13
4
votes
1
answer
240
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Characterisation of paracompact spaces by some sort of embeddability?
This question was inspired by this question.
Before I start, I don't really mean embedding in what follows. I'm tempted to use plongement, for an exotic touch, but well, that's just a rose by another ...
- 35.5k
1
vote
1
answer
83
views
Example of a collection of metacompact spaces with non-metacompact box-product
Is there an example of a family $(X_i)_{i\in I}$ of metacompact spaces, such that their box product $\prod_{i\in I}^{\textrm{Box}}X_i$ is not metacompact?
- 48.9k
12
votes
1
answer
1k
views
(Closures of sets of) operations in topological groups.
Let $G$ be a topological group. For each $n \in \mathbb{Z}$, consider the continuous functions $f_{n} \colon G \to G : x \mapsto x^{n}$, and set $F := \{f_{n} \mid n \in \mathbb{Z}\}$.
Is there a ...
- 1,498
0
votes
1
answer
76
views
Countable, $T_1$, and not metacompact
Is there a countable space that is $T_1$ and not metacompact? (A space $(X,\tau)$ is not metacompact iff there is on open cover $\cal{U}_0$ such that for every open refinement $\cal V$ there is $x\in ...
- 48.9k
-1
votes
1
answer
88
views
Interval topology on $(\mathbb{N}^\mathbb{N},\leq^*)$
Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by
$$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$
where $\downarrow x = \{y\in Q: y\leq ...
- 48.9k
1
vote
0
answers
233
views
Sum-epimorphisms and prod-monomorphisms
Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
DEFINITION 1 ...
- 7,500
3
votes
1
answer
235
views
Contractibility of connected holomorphic dynamics?
Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set :
$$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow_{k \to \infty} \infty \} $$
Question : If $K(f)$ is ...
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
votes
1
answer
284
views
Creating topological spaces with portals [closed]
I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension.
I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
- 19
2
votes
0
answers
128
views
Divisible fundamental group [duplicate]
I apologize if this question seems trivial or elementary. Is there any concrete topological space with divisible fundamental group? For example, is there any such a space the fundamental group in ...
- 2,233
5
votes
0
answers
138
views
Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?
Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...
- 1,145
2
votes
0
answers
183
views
Monadicity of profinite algebras
We can show that the category of profinite algebras, cofiltered limits of finite algebras, is monadic over Stone spaces as follows. So, I wonder if there are any other examples.
In case that I was ...
- 121
3
votes
0
answers
373
views
Closed Graph Theorem and Spaces Of Continuous Functions
Let $X$ be a (Tychonoff) topological space. Consider $C\left(X\right)$ being a topological vector space of all continuous scalar-valued functions with the compact-open topology.
Assume that $Y$ is a ...
- 5,529
0
votes
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
2
votes
1
answer
414
views
When is a space of probability measures not perfectly normal?
I am looking for examples of pairs ($(\Omega,\Sigma)$, ($\mathcal P(\Omega)$, $\tau$)), where $(\Omega,\Sigma)$ is a measurable space and ($\mathcal P(\Omega)$, $\tau$) is a space of probability ...
- 23
0
votes
1
answer
55
views
On 1-iso maps and subsets of the unit circle
Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A bijective map $\phi : S\longrightarrow S$ is called 1-iso if the following ...
- 97
1
vote
1
answer
105
views
Open cover not containing a certain subcover
Is there an infinite topological space $(X,\tau)$ with the following property?
There is an open cover ${\cal U}^*$ such that
$X\notin {\cal U}^*$;
every finite subset $F\subseteq X$ is contained in ...
- 48.9k
3
votes
3
answers
699
views
Is there a name for this property of a topology?
This property seems like it should have a nice name, but I can't find one anywhere. Does anyone know a name for this?
For each non-empty open set $U$, there exist proper open subsets $\{U_i\}_{i\in ...
- 1,488
7
votes
1
answer
796
views
Disconnecting sets
If E is a metric space, I call a subset C of E a cut if E-C is not connected and if C is minimal for this property (which is obviously equivalent to "for every p in C, E-C union p is connected". The ...
- 3,630
2
votes
1
answer
144
views
Largeness, generic, random points
As presented in Oxtoby's book ( http://link.springer.com/book/10.1007%2F978-1-4615-9964-7 ), there are two notions of largeness for subspace $Y$ of a given space $X$:
Topology: $X$ is a topological ...
- 370
3
votes
3
answers
447
views
Representations of finite commutative band semigroups
I think it's clear that commutative semigroups S that are also bands, i.e. $e^2 = e$ for all e, correspond to finite posets (consider the elements of the semigroups as sets, where the intersection of ...
- 2,216
2
votes
3
answers
614
views
A simple question on the closure of the image of a morphism
Let $X$ be a complex irreducible quasi-projective variety, $f:X\longrightarrow\mathbb{P}^N$ a morphism, $H\subset\mathbb{P}^N$ a hyperplane, $Z:=f^{-1}(H)$ which is irreducible, $Y\subset X$ a ...
- 1,159
1
vote
1
answer
106
views
Similarity graph for continuous maps between Hausdorff spaces
Let $X, Y$ be topological spaces and $f,g: X\to Y$ continuous. Then we say that $f, g$ are similar if for all $V\subseteq Y$ open we have either
$f^{-1}(V) = g^{-1}(V) = \emptyset$, or
$f^{-1}(V) \...
- 48.9k
2
votes
1
answer
403
views
The set of Upper semi-continuous functions as a ring.
I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R} $} as it's base.
If $X$ is a topological space, an upper ...
- 1,788
0
votes
1
answer
74
views
$T_2$-spaces such that the lattices of open sets can be embedded into each other
Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$.
Does this imply that $(X,\tau)\cong (Y,\sigma)$?
user62017
13
votes
0
answers
577
views
Multiplicity of ball covering
Background. My questions are motivated by the following:
A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete
and Computational Geometry, 8 (1992) 109-130) considered the ...
- 31.2k
0
votes
1
answer
501
views
2
votes
0
answers
180
views
Pro-p topology on free group
Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose ...
- 421
4
votes
1
answer
1k
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Applications of Eckmann-Hilton argument to topology
There have been a couple of posts and questions on MathOverflow about the proofs of the following two facts:
Fact 1: if $X$ is a topological space, then $\pi_k(X,x)$ is abelian for $k\ge 2$.
Fact 2: ...
- 10.9k
1
vote
1
answer
353
views
Agreement of two topologies on a linear space
I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.
Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...
- 8,512
2
votes
2
answers
1k
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When is a Hausdorff space metrisable?
This question may be a little too easy for this site, but I'll ask it anyway: when is a Hausdorff topological space metrisable?
- 29
2
votes
1
answer
432
views
Connectedness properties of groups of homeomorphisms
Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological ...
- 203
1
vote
0
answers
29
views
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
0
votes
1
answer
403
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When does a power semigroup have a zero, and what can the zero be?
Let $S$ be a semigroup. The power semigroup of $S$ is the set $P(S)=2^S\setminus\lbrace\varnothing\rbrace $ with the operation $$AB=\lbrace ab\ |\ a\in A,\ b\in B\rbrace.$$
This operation is ...
- 1,445
4
votes
1
answer
2k
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Connected level sets
This may be an ill-posed question, but suppose I have a collection of continuous, bounded, scalar-valued nonnegative functions $f_1(x,y),\dots,f_n(x.y)$ defined on the closed unit disk. Given a ...
- 645
3
votes
1
answer
148
views
Characterizing space that preserves positive-definiteness property
Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. ...
- 133
10
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2
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2k
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pro-discrete = locally compact and open normal subgroups have trivial intersection?
EDIT: After talking to some experts on the subject, I have concluded that a) the answer is not obvious or well-known for locally compact groups in general, b) the answer should be 'no' and I have some ...
- 4,728
3
votes
1
answer
4k
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A sequence with no convergent subsequence without choice
By Tychonoff Theorem $\prod_{\mathbb R} [0,1]$ is compact and since $\mathbb R=2^{\omega}$, if for $\alpha \in 2^{\omega}$, $x_n(\alpha)=\alpha(n)$ then if we consider a subsequence $x_{n_0}, x_{n_1}, ...
- 3,804
2
votes
0
answers
35
views
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