All Questions
5,185 questions
3
votes
1
answer
367
views
submonoid of a matrix monoid with a common eigenvector
Hello,
I am considering two real invertible $3\times 3$ matrices $A$ and $B$ and a nonzero vector $v\in\mathbb{R}^3$ and i am wondering if the submonoid $E$ of the monoid $(A,B)$ genererated by $A$ ...
- 69
1
vote
1
answer
636
views
Does anyone know an example of non-separable $L^1$ of a probability space?
It is easy to give examples of non-separable $L^p$ spaces by considering a measure on a big space. If one adds the condition that the space has to have total measure 1, the problem is not as easy.
...
- 96
4
votes
0
answers
98
views
Unique representability of bounded distributive lattices
Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space.
A poset $(P,\leq)$ is called (...
- 48.9k
5
votes
1
answer
1k
views
Orderings of ultrafilters
Let $U$ is a set. I will speak about filters on this set.
If $f$ is a function and $a$ is a filter then I define $f \left[ a \right]$ as
the filter whose base is $\lbrace f[A] | A \in a \rbrace$.
I ...
- 765
2
votes
1
answer
356
views
algebra-geometry duality
For topological spaces $S$ and $T$, denote by $C(S)$ and $C(T)$ the corresponding algebras of continuous real-valued functions. What are the necessary conditions that we need to impose on $S$ and $T$ ...
- 21
1
vote
3
answers
314
views
Counterpart of Weierstrass theorem
Assume that $(X,\tau)$ is a topological space and assume that every continuous mapping $f$ of $X$ into real line $\mathbb{R}$ achieves its maximum. Under which conditions on $\tau$, the space $X$ is ...
- 303
2
votes
2
answers
704
views
Topology on Set of Prime Filters of a Distributive Lattice
Given a distributive lattice $A$ we can look at $Spec(A)$, whose points are prime ideals and its open sets are given similarly to the Zariski topology on Spec of a ring. That is, the basis of open ...
- 10.5k
4
votes
1
answer
1k
views
Cantor set and Hilbert cube, or anything else?
I have recently rediscovered (after several years) the wonder of the Cantor set (so rich and so beautiful!). I have two questions that are unrelated, but they are both about Cantor sets.
Let $K$ be a ...
- 2,829
1
vote
1
answer
171
views
US does not imply AB
We say that a topological space $X$ is:
$AB$, provided that $X$ is $T_1$ and for each pair $(A, B)$ of compact, disjoint subsets of $X$ there is $U$, an open subset of $X$, such that either $A \...
- 11
6
votes
2
answers
1k
views
Computations in Knot Homology Theories
1) Relative to one another, how computable are the various knot homology theories? For example, how many crossings can we allow a knot and still hope to compute its Khovanov homology, versus its knot ...
- 2,806
4
votes
4
answers
599
views
A question about indecomposable continua.
The term "continuum" is often used to mean a compact and connected metric space. But it is
also used in a broader sense to mean any infinite, complete, separable and connected metric
space-which is ...
- 6,215
2
votes
2
answers
180
views
a question about semigroups
Let $S$ be a semigroup and $I,J$ be two ideals of $S$. For a semilattice we know that $IJ=I\cap J$. Now the question is there a semigroup with the property $IJ=I\cap J$. thanks for your attention
- 23
5
votes
1
answer
156
views
Is There a maximal space that is a P-space?
we guess there is no maximal space which is also a P-space. Am I right? Do u know a counter example?
clarifications:
Maximal space is that space with topology $\tau$ which is maximal crowded topology ...
- 113
4
votes
1
answer
371
views
Inducing metric spaces
Let $f\colon \mathbb{R}_{\geq0} \to \mathbb{R}_{\geq0}$ be a function. We say that $f$ has the property of inducing metric spaces, whenever for all metric space $(X,d)$, $(X, f \circ d)$ is also a ...
- 283
1
vote
2
answers
171
views
Questions about knot (link) of surface in four dimension
Consider three 2-torus ($S^1*S^1$) living in four space. Can I have links of these objects, which is generalization of links of circles in 3D? If so, how can I judge whether three 2-torus are linked ...
- 293
0
votes
0
answers
173
views
Minimum regular open set containing a given set in a T0 Alexandrov topological space
What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be first-...
- 1
2
votes
0
answers
188
views
Space of continuous real-valued functions on $[0,1]^\omega$ with the weak and pointwise topology
Let $C([0,1]^\omega)$ denote the set of continuous functions $f:[0,1]^\omega \to \mathbb{R}$. We endow $C([0,1]^\omega)$ with two topologies. Let $\tau$ be the pointwise topology on $C([0,1]^\omega)$ ...
- 48.9k
3
votes
1
answer
365
views
Final topology of surjective linear map on Banach space
Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||_X)$ to vector space $Y$ and denote with $\tau_L$ the final topology on $Y$ induced by $T$.
Is $\tau_L$ equivalent ...
- 315
3
votes
1
answer
896
views
Geometry Realization of Homology Class
Hello!
My question is about the realization of homology class.
The definition of the realizaion of homology class is: for manifold M and a homology class $z\in H_k(M)$, k is an integer. If we find a ...
- 703
3
votes
2
answers
340
views
What, if anything, can be said about continuous images of densely ordered spaces?
If a set is equipped with a dense, complete order then the corresponding topological space is connected - and hence, so are its continuous images, even in unordered spaces. What happens if we remove ...
- 3,612
1
vote
0
answers
260
views
Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
- 11
1
vote
1
answer
149
views
Problem about the existence of a continuous surjective map [closed]
Let $F$ be a closed set in $\Bbb R^2$, $F\neq \varnothing,\Bbb R^2$, and $F^\circ\neq \varnothing$,
does there exist a continuous surjective map from $\Bbb R\times \Bbb Z$ to $F^{\circ-}-F^\circ$？
1
vote
0
answers
81
views
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 ...
- 123
4
votes
2
answers
426
views
An approximate infinite-dimensional fixed point theorem
Given $\epsilon > 0$ and $f : [0, 1]^{\omega} \rightarrow [0, 1]^{\omega}$, can we find $x$ such that $x \in \textrm{Conv}\left( \left\{f(y) : ||y - x||_{\infty} < \epsilon\right\}\right)$?
In ...
- 777
2
votes
0
answers
139
views
Goldie's Theorem for Semigroups
Goldie's theorem is a theorem in noncommutative ring theory that gives a clear picture of semiprime Noetherian rings (actually a slightly broader class). Let $R$ be a semiprime Noetherian ring. The ...
- 6,870
-1
votes
1
answer
669
views
Stone-Cech compatification and ultrafilter [closed]
I have been studding about compatification of a topological space $X$. But I have low understanding about the Stone-Cech compatification, specially construction of the Stone-Cech compatification on ...
- 147
5
votes
2
answers
257
views
Quotients of Cantor cubes onto spaces
Let $\lambda$ be an infinite cardinal. Consider the Cantor cube $\Delta_\lambda = \{0,1\}^\lambda$. It is a standard fact in topology that the topological weight (= minimal cardinality for a basis) of ...
- 387
6
votes
0
answers
218
views
Are there any known ``topological" invariants for branched coverings?
My question is the following: let $f:\Omega\to \mathbb{R}^n$ be a branched covering, namely $f$ is continuous, discrete (each fiber is a discrete subset of $\Omega$) and open (open sets get mapped ...
- 1,881
2
votes
0
answers
343
views
continuity with respect to weak-${\ast}$ topology
Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...
- 1,835
7
votes
2
answers
370
views
Wants: Polynomial Time Algorithm for Decomposing a Multiset of Rationals into Two Additive Subsets.
First, allow me to say that this problem was posed to me by a professor in the department. It is related to his research in a way that I do not know. However, since I couldn't come up with anything ...
- 4,842
3
votes
1
answer
1k
views
$\Delta_{2}^{1}$-hard set?
Hello everybody!
I'm interested in $\Delta_{2}^{1}$ subsets of Polish spaces, i.e. those sets that are both $\Pi_{2}^{1}$ and $\Sigma_{2}^{1}$ in the boldface hierarchy of Polish spaces.
There is a ...
user11618
1
vote
0
answers
90
views
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 $\mathscr{P}_1$ (...
- 359
2
votes
1
answer
796
views
Commutative, idempotent partially ordered monoids
A unital quantale is a suplattice with a compatible monoid structure. A quantale is called idempotent if it is idempotent as a monoid (every element is idempotent) (analogously for commutativity). ...
- 2,442
5
votes
2
answers
878
views
What is an example of a non-regular, totally path-disconnected Hausdorff space?
I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for ...
- 35.5k
4
votes
1
answer
405
views
Vocabulary on monoid periodicity
I'm learning about periodic languages, and I'm confused over the vocabulary used to describe the periodicity of (syntactic) monoids.
If I understand correctly, a monoid M is periodic if :
$$(\forall ...
- 786
4
votes
1
answer
384
views
Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$
Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda $ is a hyperbolic set for $f$ when there ...
user11178
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
4
votes
1
answer
333
views
n-simplex in an intersection of n balls
Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
- 195
3
votes
2
answers
326
views
continuous R^2xR^2xR^2/E^+(2) -> R^3 injection?
This is a question that comes from my (biological) research. I'm very weak in topology, so I'm not able to assure myself of the answer. The problem is this: I'm watching an animal move in two ...
- 155
9
votes
1
answer
471
views
When do sheaves which are constant along the fibers come from the base?
Assume we are given a map $f: X \rightarrow Y$ between topological spaces which is open, surjective and has (pathwise) connected fibers. Consider categories $\text{Sh}(X)$, $\text{Sh}(Y)$ of sheaves (...
- 1,704
5
votes
1
answer
117
views
metrizable neighborhoods of compact subsets
This is a question about general topology:
Assume we are given a first countable Hausdorff space and a compact subset K.
Is it possible to find a countable basis of open neighborhoods of K ?
...
- 987
10
votes
1
answer
869
views
Completeness of Borel measure
Let $X$ be a compact Hausdorff space and $\mu$ a finite Borel measure without atoms which is outer regular with respect to open sets and inner regular with respect to compact sets. Can such measure be ...
- 277
5
votes
1
answer
381
views
Ring of a Spectral Space
It is said, as far as I can tell that an arbitrary spectral space, i.e. a space that is $T_0$, sober and quasi-compact whose collection of quasi-compact open sets forms a basis and is closed under ...
- 10.5k
5
votes
0
answers
195
views
Inverse limit in shape theory
Is the shape theory of Hausdorff compact spaces complete with respect to the inverse limit operation?--complete means that for every inverse system of Hausdorff compact spaces, and the shape morphisms ...
- 7,500
5
votes
1
answer
214
views
Which combinations of normality, separability, and paracompactness do complex manifolds possess?
I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have.
Is there a non-separable complex manifold? Can a non-separable complex ...
0
votes
0
answers
85
views
Right split for homomorphism onto $S_\infty$
Let $G$ be a closed subgroup of $S_\infty$ and let $f:G\rightarrow S_\infty$ be a continuous surjective homomorphism. Under which conditions $f$ has a right split, i.e. there exists some $g:S_\infty\...
- 2,882
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
2
votes
1
answer
296
views
Does every ultrafilter has single limit imply Hausdorff separation
If a topological space $X$ enjoys the property that every ultrafilter $U$ on $X$ has a single limit, must $X$ be a Hausdorff space?
(Ultrafilters here consist of arbitrary subsets (so not necessarily,...
- 427
2
votes
0
answers
203
views
Profinite Topology
Let $V$ and $W$ be pseudovarieties of finite groups. For a finite inverse monoid $M$, the $V$-kernel of $M$ is defined to be the intersection of all sets $f^{−1}(1)$, $f$ is a relational morphism ...
- 421
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