All Questions
5,184 questions
3
votes
1
answer
149
views
Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods
Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?
- 1,145
6
votes
1
answer
765
views
Are finite colimits of topological spaces stable under pull-back?
The category of topological spaces has a forgetful functor to set which commutes with both small limits and colimits (it has both a left and a right adjoint). Moreover Set is a Grothendieck topos and ...
- 27.5k
6
votes
2
answers
492
views
Distinct, non-homeomorphic, profinite topologies on a given abstract group ?
Just a silly little question which arose in connection with infinite Galois groups and their Krull topology:- can a given abstract group be endowed with distinct, non-homeomorphic, profinite ...
4
votes
0
answers
146
views
A question on extension of $Z^{*}$ algebras
A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence of ...
- 356
1
vote
1
answer
2k
views
Covering the Rationals -- A Paradox? [closed]
Covering the Rationals -- A Paradox?
The following seems to yield a paradox. Can anyone provide the proper resolution?
Preamble
It is easy to show that between any two reals there is a rational. If ...
8
votes
1
answer
768
views
What information can one recover from the induced map on homology?
The following question came up while constructing delay embeddings of time series data.
Consider an unknown topological space $X$ and an unknown continuous function $f:X \to X$. We are given a ...
- 15.5k
0
votes
1
answer
851
views
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
1
vote
1
answer
310
views
A question from Arhangel'skii-Buzyakova
The question is also posted here, however there is no answer.
Recently, I am reading the paper: On linearly Lindelöf and strongly discretely Lindelöf spaces by Arhangel'skii and Buzyakova. Here is ...
- 654
12
votes
1
answer
1k
views
Fixed point theorems and equiangular lines
I've been thinking about the equiangular lines (or SIC-POVM) conjecture, and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking ...
- 6,342
5
votes
2
answers
709
views
profinite spaces are the pro-completion of finite sets
The title sounds tautological, right? Perhaps I'm missing something completely trivial here ...
Assume $X$ is a compact totally disconnected hausdorff space. It is known that $X$ can be written as ...
- 63.1k
10
votes
0
answers
744
views
Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?
Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...
- 888
2
votes
1
answer
297
views
A fixed point problem
Let $A = \lbrace (tr,1-t)\; | \; t \in [0,1], r \in \Bbb{Q}\rbrace$. Is it true that any continuous function from $A$ into $A$ has a fixed point?
- 191
2
votes
2
answers
1k
views
Simple question of topological cofibration
I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the ...
- 367
4
votes
1
answer
425
views
Ring structrures on R^n
Consider a commutative ring $A= ( \mathbb{R}^n , + , \times) $, where $+$ is the usual one. Assume further that $ \times $ is continuous (with respect to the usual topology). Let $H$ be the set of non ...
- 7,249
4
votes
0
answers
326
views
Sequences and pseudocharacter in compact spaces
Is there a consistent example of a compact Hausdorff space $X$ on which the following holds?
i) there is a $Y \in {[X]}^{\aleph_1}$ such that $\psi (Y) = \aleph_1$; and
ii) there is no non-trivial ...
- 627
0
votes
1
answer
396
views
A Question about SO(n)
My question is:
How to find out all the finite subgroup of SO(n)? Or just for the simple case SO(4) SO(5)?
With more discribe:
If $S^n\backslash \Gamma$ is a manifold,
I just want to know that ...
- 703
6
votes
0
answers
561
views
Continuous images of Cantor cubes
The original title of this question was "Is there only one (up to homeomorphism) zero-dimensional homogeneous dyadic space of weight $\mathfrak{c}$?". I changed it with the hope of getting a bit more ...
- 11.5k
5
votes
1
answer
2k
views
How come nowhere dense subsets implies discrete?
Hi, I am reading an article and have encountered a remark in a proof which is not clear to me.
Maybe someone can help?
The proposition is:
Let X be a topological space without isolated points having ...
- 89
8
votes
0
answers
103
views
Locales satisfying DC?
Is there a nice (topological) characterization of the locales such that the axiom of dependant choices holds in the internal logic of the topos of sheaves ? I would also be interested in the case of ...
- 42.4k
6
votes
0
answers
243
views
A compactification of the non-negative rationals with the discrete topology
Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is ...
- 25.8k
1
vote
1
answer
400
views
$G_\delta$-diagonal
Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't
a $G_\delta$-diagonal? A topology space has a $G_\delta$-diagonal when there is a sequence
${G_n}$ of ...
- 654
6
votes
1
answer
396
views
Does a metric refine the weak-* topology on a dual space?
Let $X$ be a topological affine space over $\mathbb C$, with no additional assumptions. Let $X^*$ denote its dual space of continuous affine functionals $X \to \mathbb C$, equipped with the weak-$*$ ...
- 8,512
0
votes
1
answer
114
views
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
3
votes
0
answers
115
views
Cardinality based results in Topological Vector Spaces?
Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen:
TVS's $V$ of larger ...
- 437
5
votes
1
answer
968
views
Can topologies induce a metric? (revised)
This is a revised version of a question I already posted, but which patently was ill posed. Please give me another try.
For comparison's sake, the axioms of a metric:
Axiom A1: $(\forall x)\ d(x,x) =...
- 9,720
2
votes
0
answers
246
views
A possible generalization of the Borsuk Ulam theorem via action of symmetric groups
The symmetric group $S_{m}$ is equiped with the counting Har measure $\mu$ and the obvious sgn character. Assume that $S_{m}$ acts on $S^{n}$, $n\geq m-1$ and $f:S^{n}\to \mathbb{R}^{n}$ ...
- 356
1
vote
0
answers
138
views
Minimum rank of certain matrices
Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in $\...
- 13.9k
4
votes
1
answer
191
views
Progress on group languages characterizations
Def. A group language is a recognizable language whose syntactic monoid is a group.
q1. Is it known a "nice" combinatorial characterization of group languages ?
q1.1. If no, is it well understood ...
- 424
5
votes
1
answer
378
views
Representations of products of groups (and monoids)
I have very little knowledge of representation theory, but the following has come up in my summer undergrad research project (relates to conformal field theory and geometric function theory).
Suppose ...
- 51
4
votes
1
answer
2k
views
Closed connected subset of a connected set
Let $A$ be a closed set and let $B$ be a connected set such that $A \subset B$.
Does there always exist a closed connected subset $C$ of $B$ that contains $A$?
What if $B$ is path connected, is ...
- 230
5
votes
0
answers
265
views
Quotienting disk inside sphere result in sphere
Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where $S^...
- 2,023
3
votes
0
answers
113
views
Adjunctions of uniformly locally connected spaces
A space $X$ is uniformly locally connected (ULC) if there exists an open neighbourhood $U$ of the diagonal $\vartriangle_X$ in $X \times X$ and a homotopy $H: U \times I \to X$ between $\pi_1|U$ and $\...
- 81
3
votes
1
answer
251
views
In which cases a fiber bundle over a circle is a graph-manifold?
A fiber bundle over a circle $M^{3} \longrightarrow S^{1}$ with fiber a surface $F_{g}$ is characterized via a homeomorphism $\varphi \colon F_{g} \to F_{g}$. It can be one of the following: periodic, ...
- 192
13
votes
1
answer
719
views
Homotopy theory for spanning trees of a graph
I am studying a paper of L. Lovász, ``A homology theory for spanning trees of a graph,'' but professor Babai has told me that Lovász later realized that this work is better framed in the language of ...
- 5,898
5
votes
1
answer
201
views
A question on minimal idempotent ultrafilter on N^2
Is there some minimal idempotent ultrafilter $q \in \beta( \mathbb{N}^2)$ (with respect to the law $"+"$) such that any $A \in q$ is a subset of $\mathbb{N} \times \{ 0 \} $ ?
(See for example http:/...
- 7,249
7
votes
1
answer
789
views
Counting submanifolds of the plane
After thinking about this question and reading this one I am led to ask for an uncountable collection of homeomorphism types of boundaryless connected path-connected submanifolds of the plane.
My ...
- 28.2k
4
votes
0
answers
349
views
Why does $\beta \mathbb{R} \setminus \mathbb{R}$ have exactly 2 connected components? [closed]
Whilst reading about extensions of C*-alegbras, this topological fact was stated. I understand why $\beta \mathbb{R} \setminus \mathbb{R}$ has at least 2 connected components (it surjects onto the two ...
- 101
5
votes
1
answer
977
views
How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?
How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface ...
- 1,499
7
votes
1
answer
650
views
Cones, monoids, and the space of (very) ample divisors
An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...
- 1,871
4
votes
1
answer
297
views
Reference for subsemigroups of $\mathbb{N}^n$
A well known result about the natural numbers $\mathbb{N}$ says that for any finite subset $A \subset \mathbb{N}$ there exists $R \ge 0$ such that if $n$ is in the subgroup of $\mathbb{Z}$ generated ...
- 15.4k
4
votes
1
answer
224
views
Bases of open sets with connected intersections
I'm interested in knowing classes of topological spaces $X$ which admit a basis of open sets $\{U_i\}_{i\in I}$ such that $U_i\cap U_j$ is connected for all $i,j\in I$. Do manifolds have this property?...
- 15.2k
3
votes
0
answers
551
views
Is the limit set of a group action always closed?
Let $G$ be a discrete group acting on a compact metric space $X$. A point $x\in X$ is called a limit point, if there is a base point $x_0\in X$ and an injective sequence $(x_k)_{k\in\mathbb{N}}$ in ...
- 2,581
2
votes
1
answer
800
views
Zero dimensional iff every closed set is a retract.
In the Kechris book on (Classical) Descriptive Set Theory there is a claim that a separable metrisable space is zero dimensional if and only if every closed set is a cts retract of the whole space (...
- 21
2
votes
0
answers
119
views
Are all locally compact anisotropic groupoids etale up to equivalence?
By groupoid I mean "open topological groupoid",i.e. topological groupoids whose source and target maps are open surjections, and the notion of equivalence I'm considering is the isomorphism in the ...
- 42.4k
6
votes
2
answers
657
views
Properties of the class of topological spaces possessing a CW-structure
Let ${\mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure).
Is it true that for a covering map $E\stackrel{f}{\to} ...
- 2,756
9
votes
1
answer
726
views
Uniform Embedding into Euclidean Space
Given a locally compact, separable, metric space $X$.
When does $X$ uniformly embed into some Euclidean space?
This means, when does there exist some integer $n$ and a closed subset $Y\subset\...
- 3,497
6
votes
2
answers
462
views
need references regarding the elementary theory of free semigroup and free abelian groups
Recently, I read that two free abelian groups $S$ and $T$ have the same elementary theory if and only if rank$S$=rank$T$. Does anyone have a reference with a proof of this? Also, what is known about ...
- 549
6
votes
3
answers
582
views
profinite spaces coming from profinite groups
This is probably well-known:
Does every nonempty profinite space occur as the underlying space of a profinite group? If not, which conditions have to be imposed?
- Is every profinite group ...
- 63.1k
3
votes
1
answer
364
views
Existence of a non-submetrizable topological space $(X, \tau)$
We recall that a topological space $(X,\tau)$ is submetrizable, if there is a coarser metrizable topology $\tau'$ that $\tau\supseteq\tau'$.
one of the properties of these topological spaces is ...
- 1,788
3
votes
2
answers
384
views
ED compact $K$ such that $C(K)$ is not a dual Banach space
Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not hyperstonean). ...
- 55