All Questions
5,185 questions
3
votes
1
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860
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Counterexemple to Urysohn's lemma in a topos without denombrable choice ?
Hello !
The Urysohn's Lemma assert that in every topological spaces which is normal two closed subset may be separated by a real valued function. It's proof use axiom of countable choice (but not the ...
- 42.4k
2
votes
0
answers
139
views
Centralizer of a dense subgroup in a maximal subgroup of a reductive group
I am looking for a reference to the following statement
"Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...
- 21
5
votes
1
answer
247
views
Maps between spaces of non-empty compact subsets with the Hausdorff distance (reference request)
Let $X, Y$ be metric spaces, and let $PX$ (resp. $PY$) be the set of all non-empty compact subsets of $X$ (resp. $Y$) with the Hausdorff metric. A continuous map $f\colon X\to Y$ induces a continuous ...
- 666
6
votes
1
answer
1k
views
Some questions on Nicolai Reshetikhin's lectures on quantization of gauge theories.
This in reference to this fascinating lecture by Nicolai Reshetikhin-
http://staff.science.uva.nl/~nresheti/Holb-Quant-Gauge.pdf
Given what is said on page 13 in section 4.1 its not clear to me why ...
- 3,541
4
votes
4
answers
1k
views
Boundary of planar region
Is there a necessary and sufficient condition for the boundary of a planar region to be a finite union of Jordan curves?
- 1,169
2
votes
1
answer
285
views
Unbounded convex not containing a ray - example without using a basis
I prove here that an unbounded convex in a finite dimensional space contains a ray. At the same place, I give an example of an unbounded convex not containing a ray in the case of an infinite ...
- 1,355
2
votes
0
answers
261
views
Normed space that is sigma-totally-bounded but is not sigma-compact
Q1: Is there a separable normed space that is not sigma-compact, but is a countable union of
totally bounded closed subsets?
A test case is the space $C^1(I)$ with the $C^0$ norm where $I=[0,1]$. ...
- 29.1k
5
votes
0
answers
219
views
Topological Subset Take-Away
David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose proper nonempty subsets of $S$ such that if a subset is chosen, then none ...
user62675
1
vote
1
answer
254
views
Interval topology and order convergence topology
Throughout this post, let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\...
user62017
5
votes
1
answer
223
views
A realcompact analogue of the Baire category theorem
Let $\frak{m}$ be the least measurable cardinal. A space $X$ is realcompact if it is homeomorphic to a closed subset of some product $\mathbb{R}^I$. Let $X$ be realcompact with $P_\frak{m}$ topology, ...
- 3,041
2
votes
2
answers
328
views
non-P-points a Baire space
Let $X$ be a compact Hausdorff space. A P-point in $X$ is a point which does not lie in the boundary of the cozero set of a continuous real-valued function on $X$.
Question. Suppose that $X$ has no ...
- 607
4
votes
1
answer
582
views
Density of linear functionals in $L^2$
Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals $...
- 8,512
3
votes
0
answers
83
views
Is the increasing union of disk bundles a disk bundle?
Setup: Let $B$ be a $C^r$ $n$-manifold ($r \geq 1$) and $M$ a closed $k$-dimensional $C^r$ submanifold of $B$. Assume there exists a smooth retraction $p:B \to M$ which is also a submersion, so that $...
- 501
1
vote
0
answers
96
views
Induced structure of topological group [closed]
If we consider a closed Jordan curve $\mathcal{C}$, I know that it's homeomorphic to the circle $S^1$. Now I take an homeomorphism $\phi:S^1\longrightarrow\mathcal{C}$ and this homeomorphism induces a ...
- 1,252
2
votes
1
answer
557
views
Is this a closed set?
Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...
- 8,512
3
votes
1
answer
127
views
Construct a specific base for Fine uniformities in the diagonal(Entourages) case
For every uniformizable space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity.
To construct Fine uniformities, Let ...
- 153
2
votes
0
answers
160
views
Pre-cosheaf of connected components
Consider a continous map $f:Y \to X$ between topological spaces. The pre-cosheaf $\mathcal{F}: Open(X) \to Set$ of connected components of the inverse image is defined as $\mathcal{F}(U):= \pi_0(f^{-1}...
- 229
-1
votes
1
answer
73
views
existence of continuous functions with values in the fiber of a closed bundle
Let $ A \subseteq \mathbf{R}^{n} $ be a closed set and let $ B $ be a closed unit normal bundle over $ A $ ( that means for every $ a \in A $ we have closed subset $ B_{a} \subseteq \mathbf{S}^{n-1} $ ...
- 541
4
votes
2
answers
414
views
Is it impossible for the dimension of a topological space to increase under a smooth map?
First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots ...
- 3,245
24
votes
0
answers
2k
views
Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property, without Choice
While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...
user5810
1
vote
0
answers
56
views
A topology for which symplectic forms are dense in skew forms
Let $V$ be a vector space over an algebraically closed field. Let $S$ denote the vector space of skew-symmetric bilinear forms on $V$. When $V$ is finite dimensional the subset of $S$ consisting of ...
- 599
1
vote
0
answers
50
views
Closedness of the range of the distorsion of the multiplicative monoid of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is an element $x \in H \setminus H^\times$ such that $a \ne xy$ for all $x, y \in H \setminus H^\times$, where $H^\...
- 10.5k
5
votes
1
answer
494
views
When is a $*$-homomorphism between multiplier algebras strictly continuous?
(This question was posted on MSE here but didn't get any answers.)
The strict topology on the multiplier algebra M(A) of a C*-algebra A is that generated by the seminorms
$$ x\mapsto \|ax\|\quad x\...
- 1,336
2
votes
1
answer
258
views
Topological degree of homogeneous function of degree k [closed]
Let $F:\mathbb{C}\to \mathbb{C}$ be a homogeneous map of degree $k$ (i.e., $F(tx)=t^kF(x)$, $t>0$). It is true that $F$ has topological degree less than or equal to k?
This is true if F is ...
- 81
0
votes
1
answer
149
views
Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?
Is there somone help me to show that if this problem have positive Answer :
Problem :Can every non-discrete topological group G be algebraically gen-
erated by a nowhere dense subset ?
Thank ...
7
votes
1
answer
722
views
How is called a semigroup...
Does anyone know, how is called a semigroup in which every equation $ax=b$ has only a finite set (maybe empty) of solutions?
- 3,110
2
votes
1
answer
171
views
Closed sets in ordinal spaces
I'm studying the ordinal space $[0,\kappa[$ where $\kappa\neq \omega$ is a cardinal of countable cofinality and I want to know why there are in $[0,\kappa[$ two disjoint closed sets of cardinality $\...
- 225
2
votes
3
answers
501
views
Generalized free product of semigroups with amalgamated subsemigroups
Hanna Neumann in
[American Journal of Mathematics, 1948,
http://www.jstor.org/discover/10.2307/2372201?uid=2&uid=4&sid=21102497379451 ]
introduced a notion of generalized free product of ...
- 3,110
6
votes
1
answer
119
views
Universality with respect to quotients
Is there an infinite cardinal $\kappa$ for which the following statement (S) true?
(S) : There is a topology $\tau_\kappa$ on $\kappa$ such that for all topological spaces $(X,\tau)$ with $|X|\leq \...
- 48.9k
4
votes
2
answers
273
views
Question about lower homology class of cobordism
Assume there are three differential oriented manifold $M_0$, $M_1$, $W$ with $\partial W= M_0 \coprod -M_1$. Denote dim $M_0$=dim $M_1$=n, and dim W=n+1.
We know that for the highest homology class, ...
- 703
4
votes
1
answer
162
views
Some questions about "inspecting" the boundary of a closed ball in Hilbert space
Let H be a separable Hilbert space and suppose that H is infinite dimensional. Let B be a closed ball of H-which has a positive radius-and let S be the boundary of B. A non-empty subset C of H is an "...
- 6,215
1
vote
1
answer
582
views
T2 ⇒ KC ⇒ US ⇒ T1
In a topological KC-space, every compact space is closed.
In a US-space, each convergent sequence has a unique limit.
So, T2 ⇒ KC ⇒ US ⇒ T1, but the converse implications do not hold.
(a): Can ...
- 147
3
votes
1
answer
184
views
What do sparse sets in a norm topology look like in the weak* topology?
I'm wondering if a very "sparse" set in a normed vector space can look connected in the weak* topology. Specifically,
Let V be a Banach space, V* its dual, and X a (uncountable) subset of the unit ...
- 333
2
votes
1
answer
637
views
Topological properties of SpecMax(A)
We consider $A = C_{b}(X)$, the ring of continuous bounded functions on a completely regular space $X$. Let $\DeclareMathOperator{\SpecMax}{SpecMax} \SpecMax(A)$ be the set of maximal ideals of $A$ ...
- 933
3
votes
1
answer
792
views
A closed connected component in a topological space does not contain any path-connected subset?
Does there exist such a non-trivial closed connected component U of some connected topological space X or a non-trivial connected topological space X that do not contain any non-trivial path-connected ...
- 1,881
1
vote
2
answers
1k
views
Hypercohomology of a complex of sheaves that might be acyclic (or might not)
Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves
...
- 360
3
votes
1
answer
143
views
What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite
When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...
- 1,060
6
votes
0
answers
322
views
Terminology for notion dual to "support"
If $X$ is a set (feel free to think of it as finite, but it doesn't have to be) and $f$ a real function on $X$, call the support $\operatorname{supp} f$ the subset of $X$ consisting of all elements $i\...
- 21.5k
1
vote
2
answers
406
views
Understanding the left-separated spaces
A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$.
Could someone post some left-separated space to help me understand such ...
- 654
5
votes
0
answers
207
views
Can the compactification of a (co)tangent bundle equipped with Saski metric be viewed as a "Wick rotation"?
We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like
\...
- 51
2
votes
1
answer
247
views
Rep of Non-Commutative Monoids
Let M be a non-commutative monoid. It is possible that all representation of M are one dimensional ??
(for groups the answer is negative. Take a non zero x=[a,b]. Take a representation where x does ...
- 2,384
0
votes
0
answers
331
views
Idempotent ideal in ring of continuous functions
Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be be an idempotent ideal?
- 1
1
vote
0
answers
47
views
Is the minimality of complete topological groups recognizable by closed separable subgroups?
A topological group is called minimal if it admits no strictly weaker Hausdorff group topology.
By Prodanov-Stoyanov Theorem, a complete Abelian topological group is minimal if and only if it is ...
- 42k
2
votes
0
answers
62
views
The topology of the space of simple tensors [duplicate]
We identify the vector space tensor product $\mathbb{R}^{m} \otimes \mathbb{R}^{n}$ with $\mathbb{R}^{mn}$
Let $X$ be the space of all non zero simple tensors $X=\{a\otimes b \mid a\in \...
- 366
3
votes
0
answers
402
views
Generalization of Jordan Curve Theorem
Jordan Curve Theorem says that any plane continuum homeomorphic to $\mathbb{S}^1$ separates the plane into exactly two components.
Now
"Let $\alpha$ and $\beta$ be two homeomorphic plane continua. ...
- 615
7
votes
1
answer
593
views
Question about topological monoid maps
Let Mon be the category of topological monoids. I am happy to work with the model structure mentioned here:
Model Structure/Homotopy Pushouts in topological monoids?.
I'm looking for a reference ...
- 18.9k
2
votes
0
answers
224
views
cross-sections of a sphere bundle
Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...
- 2,223
0
votes
0
answers
125
views
Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a fixed finite set and $x\in X$
Preliminaries.
Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set
$$
X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\},
$$
which is just the usual ...
- 10.5k
1
vote
0
answers
58
views
A class of finitely generated semigroups
Let $G$ be a finitely generated group with a probability measure $\nu$. Suppose we have a finite first moment function $f:G\to \mathbb{R}$, i.e, such that $\Sigma_{g\in G}f(g)\nu(g)< \infty$. Then, ...
- 49
-1
votes
1
answer
148
views
Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]
Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...
