All Questions
5,184 questions
4
votes
1
answer
2k
views
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
4
votes
1
answer
234
views
Statistical models in terms of families of random variables
A statistical model is a function $P : \Theta \to \Delta(X)$, where $\Theta$ is a parameter space, and $\Delta(X)$ is the set of probability measures on a state space $X$.
Suppose that $\Theta$ and $...
- 8,512
-4
votes
1
answer
483
views
Why $z \in \overline{A}$? [closed]
In the Picture blew:
The paper can be downloaded here. Why $z \in \overline{A}$?
Thanks.
A point $x$ of a space $X$ is called $G_\omega$-separated from a subset $Y$ of $X$ if there is a closed $G_\...
- 654
10
votes
2
answers
2k
views
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
1
vote
1
answer
189
views
subspace in pseudotopological space
Every topological space gives rise to a pseudotopological space. Conversely, if $X$ is a pseudotopological space then we can define a topology on $X$ such that every filter converging to $x$ in the ...
- 41
0
votes
0
answers
238
views
Pro-constructible subset of scheme intersects very dense subsets?
Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$?
If $Y$ is just constructible, this is true.
...
2
votes
1
answer
1k
views
Spectral sequences in Hypercohomology of sheaves
Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this ...
- 360
13
votes
1
answer
736
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,547
1
vote
2
answers
443
views
submonoids of Z_n
Anyone knows how to describe explicitly the submonoids of Z_n, regarded as a multiplicative
monoid?
3
votes
0
answers
303
views
Pseudomodules, "general coherence theorem"
A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...
- 2,312
3
votes
0
answers
201
views
Which compact topological spaces are homeomorphic to their ultrapower?
It is well known that for any compact metric space $(X, d)$, and any ultrafilter $\mu$ there is a map $i_\mu:\prod_\mu (X, d) \to (X_d)$ in the category of metric spaces and Lipschitz maps where $i_\...
- 2,221
3
votes
1
answer
99
views
cartesian product rigidity for the punctured open disc
Q1: Let $D^n$ ($n\geq 1$) be the n-dimensional open disk. If $D^n-\{0\}$ is homeomorphic to $X\times (0,1)$, for some topological space $X$, does it necessarily follow that $X$ is homeomorphic to $S^{...
- 7,609
4
votes
1
answer
122
views
Approximation of sets by sets with regular border
What kind of conditions on a (bounded) set $E \subset \mathbb{R}^{n}$ ensure that it can be approximated from outside/inside by sets with regular border (say Lipshitz or $C^{k}$ conditions) in the ...
- 321
12
votes
2
answers
2k
views
Topological Rings
Is it true that, if S is a subring of a separable topological Noetherian ring R,
then S is separable, too ?
- 4,060
10
votes
2
answers
752
views
Adding a formal inverse of an element to a free monoid
Let $FM_2=\langle a,b\rangle$ be the free monoid of rank 2. If we add a formal inverse to the word $aba$, we get the free group $F_2$ (because both $a$ and $b$ will have inverses).
Question: For ...
user6976
1
vote
1
answer
121
views
A Hausdorff atom in lattice of group topologies
Do you have an example of an infinite Hausdorff nonabelian topological group $(G,\mathcal T)$ such that for any nontrivial group topology $\mathcal S$ on $G$ with $\mathcal S\subseteq \mathcal T$ we ...
- 1,145
2
votes
1
answer
689
views
Partitions of an interval
This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there.
Specifically, consider "partitions" ...
- 2,358
2
votes
2
answers
241
views
If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...
- 10.5k
6
votes
2
answers
1k
views
Quantitative questions about the size of a finite epsilon net
Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...
- 943
4
votes
1
answer
2k
views
A question about group action on topological space
Let $G$ be group and let $X$ be a topological space on which $G$ acts continuously. Now let us consider the following two properties relative to the group action:
(a) For every compact subset $K\...
- 7,609
5
votes
1
answer
458
views
Ideals of $C(X)$ with only finitely many number of zerosets
We denote the rings of all real valued continuous functions on compeletely regular Hausdorff space $X$ by $C(X)$.Let $I$ be a ring ideal of $C(X)$. define $$Z[I]:=\lbrace Z(f):\;f\in I\rbrace$$ where ...
- 1,788
4
votes
1
answer
1k
views
Abstract definition of properly discontinuous action
A discrete group $G$ acts properly discontinuously on a manifold $M$ if the set $\{g\in G\mid gK\cap K\neq \emptyset \}$ is finite for every compact $K\subset M$.
Is there a more abstract ...
- 1,211
5
votes
1
answer
540
views
Bitopological spaces and algebraic topology
Is it possible to introduce the concept of bitopological spaces such as $(X,T_1,T_2)$ (introduced by J.C.Kelly see Proc. London Math. Soc. (3) 13 (1963) 71–89 MR0143169, J.C. Kelly) in the homotopy ...
- 243
1
vote
1
answer
244
views
Finding a good ordering of $\mathbb{Q}$
Oftentimes in density arguments we let $\{x_n\}$ be a dense sequence and this is sufficient to imply the desired result.
From a research question I am working on I have simplified the example/...
- 882
1
vote
0
answers
127
views
Category-theoretic characterization of zero-dimensional spaces
Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
- 369
1
vote
1
answer
371
views
countably normed spaces and countably normed spaces [closed]
Why locally convex spaces are not presented as countably normed spaces i.e an infinite sequence of norms (see Generalized functions Tome 2 by Gelfand and Chilov) in the western mathematical ...
- 308
6
votes
1
answer
815
views
When is a Topological pushout also a Smooth pushout?
I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:
Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram ...
- 732
7
votes
0
answers
204
views
Is $(\omega+1)^\omega$ with the box topology ultraparacompact?
Let $\omega+1$ be endowed with the interval topology, that is $U\subseteq (\omega+1)$ is open if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite. We call $U\subseteq (\omega+1)$ basic if ...
- 48.9k
4
votes
1
answer
255
views
Forcing over the poset of nonempty open subsets of a nice topological space
Is there anything sensible to be said concerning a notion of forcing given by the poset of nonempty open subsets of the sort of topological space that comes up in ($e.g.$ algebraic) topology? If so, ...
- 2,550
1
vote
0
answers
96
views
Is there another equivalence relation on based maps between spheres which form the same graded ring as the homotopy groups?
Let $\sim$ be an equivalence relation on continuous based maps from $S^k$ to $S^n$, where $k$ and $n$ range over the positive integers.
Suppose that
Given maps $f, f^\prime: S^k \to S^n$ and $g, g^\...
user2529
3
votes
0
answers
334
views
Homeomorphism of compact Hausdorff spaces
(Note: I asked this question at MSE over a day ago and received no answer, so I'm now reposting it here. Link: https://math.stackexchange.com/questions/853500/homeomorphism-of-compact-hausdorff-spaces)...
- 191
8
votes
2
answers
689
views
What does the space induced by this unusual metric(?) on R/Z look like?
The motivation for this question comes from music theory. Dmitri
Tymoczko models "good" voice leading as minimizing distance between
pitches in successive chords. While this theory works well for ...
- 3,093
5
votes
1
answer
479
views
A question about Q?
Let A=$\{a_n : n\in \omega \}\subset 2^{\omega\times\omega}$ be nonempty countable without isolated points (i.e. homeomorphic to $\mathbb{Q}$), and satisfy $ \forall n\in \omega \exists^\infty m|\{k:...
- 427
11
votes
0
answers
584
views
A proof of the gluing axiom of a TQFT
I posted the following question on math stackexchange but I have not received any answer.
So I hope people here can help me.
In the book Lectures on tensor categories and modular functors by Bakalov ...
user22741
6
votes
1
answer
634
views
Arbitrary small positive lower semi continuous functions
This question is a generalization of the question posed in this page to lower semi continuous functions. so let me describe the Question in the following way.
Def: Let $(X,\tau)$ be a Tychonoff ...
- 1,788
2
votes
2
answers
1k
views
Are coordinate functions on topological vector spaces always continuous?
Let $V$ be a Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ such that
$\;$ for all functions $c : B\to \mathbb{K}$, if $\displaystyle\sum_{...
user5810
3
votes
1
answer
165
views
Properties of open covers
I am reading this article in which two properties of open covers are described:
$\gamma$-property: If $\mathcal U$ is an open $\omega$-cover of $X$, then there sequence $\{ G_n : G_n \in \mathcal U\} ...
- 213
2
votes
0
answers
473
views
Homeomorphisms between infinite-dimensional Banach spaces and their spheres
As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere.
Unfortunately I do not have his book but I want to know is this theorem true without ...
- 21
1
vote
1
answer
2k
views
On Zariski Dense Subsets
Can we find a Zariski-dense subset $U$ of the affine plane over the complex number field, such that any subset $U^{\prime}$ of $U$ has its closure either the whole affine plane or finite number of ...
- 211
0
votes
2
answers
909
views
Topology generated by the collection of open sets
Hello, there is a statement as following:
If every point of X is a G_delta and X is T_1, then take Y = set of X,
plus the topology generated by all open sets needed to prove G_delta-ness of every ...
- 654
3
votes
1
answer
529
views
Study of free monoids of the recursive S. Eilenberg.
Compared to the usual treatises on recursion (eg, Rogers H. "Computability and Undecidability." McGraw-Hill, New York) the book of Samuel Eilenberg & Calvin C. Elgot "Recursiveness" treats such ...
- 4,165
5
votes
2
answers
292
views
Simultaneously minimizing intersections
This may be a standard problem in homotopy theory, but I don't know a good reference.
Let $\Sigma$ be a smooth, oriented surface, and let $X_1,X_2$ and $X_3$ be three smoothly embedded curves in $\...
- 13k
2
votes
2
answers
331
views
Uniformities generated by metrics.
Any uniformity on a set $X$ is generated by a family of pseudometrics on $X$. So if $(X,\mathcal D)$ is a uniform space there's a set $P$ of pseudometrics on $X$ with
$$\mathcal D=\left< \bigcup_{...
user31967
4
votes
1
answer
314
views
A semigroup with the property that $x^n = a$ has at least one solution
Is there a standard name for a (multiplicatively-written) semigroup $(A, \cdot)$ such that, given an arbitrary $a \in A$, the equation $x^n = a$ has at least one solution $x \in A$ for each $n \in \...
- 10.5k
11
votes
1
answer
536
views
Can dividing out a group action can increase the Lebesgue dimension ?
Given any space $X$ of Lebesgue dimension at most $n$. Suppose a group $G$ acts on $X$ continuously. Can the dimension of the quotient $G\backslash X$ exceed the dimension of $X$?
I know examples, ...
- 11.1k
1
vote
0
answers
663
views
Extending a homeomorphism from a dense set [closed]
Let $X$ and $Y$ be Hausdorff topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ ...
- 8,512
3
votes
1
answer
394
views
Residual finiteness of groups versus residual finiteness of semigroups
A group $G$ is residually finite if, for any two elements $g$ and $g^\prime$ in $G$, there is a finite group $G^\prime$ and a (group) homomorphism $f: G \rightarrow G^\prime$ such that $f(g)$ doesn't ...
- 155
1
vote
0
answers
479
views
Comparing two metrics on the space of infinite sequences and relating open and closed sets
Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it
$$
d(u,v) := \frac{1}{2^r} \mbox{ with } ...
- 798
5
votes
2
answers
521
views
Freeing a sphere from within a sphere
We can embed $S^2\times I$ into $\mathbb{R}^3$ by taking a compact 3-ball and removing an open 3-ball from its interior. Taking the boundary gives an embedding $i: S^2\sqcup S^2\hookrightarrow\mathbb{...
1
vote
3
answers
995
views
SO(3) knot polynomials
Can one use the real lie algebra so(3) to get knot polynomials? If so, do they have a skein relation (I presume they would, if they come from R-matrices in some standard way. If so, is the R-matrix ...
- 1,129