All Questions
5,184 questions
0
votes
1
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91
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Intersection of complements of connected components
Let $(X,\tau)$ be a topological space, $S\subseteq X$ such that there is $x^*\in X\setminus S$.
Let $E$ be the connected component of $X\setminus S$ that contains $x^*$. Let ${\cal C}$ be the ...
- 48.9k
2
votes
2
answers
1k
views
When is the group of homeomorphisms of a compact space locally compact?
When is the group of homeomorphisms of
a compact space locally compact?
I am interested in finding out when the group of homeomorphisms of a compact topological space $X$ (with appropriate topology ...
- 1,771
4
votes
0
answers
152
views
On the computational complexity of the Hilbert polynomial of numerical semigroup rings
Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that $\ell(R/\...
- 2,773
2
votes
1
answer
187
views
Unitization via "End points compactification"
We know that every compactification of locally compact Hausdorff spaces correspond to a unitization of $C^{*}$ algebras. For example the one point compactification corresponds to the minimal ...
- 356
1
vote
1
answer
84
views
Number of continuous characters on an infinite Hausdorff precompact abelian group with exponent $p$
Let $(G,\mathcal T)$ be an infinite Hausdorff precompact abelian topological group and let $G$ have exponent $p$ where $p$ is a prime number.
Can it be proved that there are at least $p+1$ continuous ...
- 1,145
12
votes
0
answers
461
views
3 manifolds with diffeomorphic unit tangent bundles
What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?
- 131
3
votes
3
answers
2k
views
motivation for compactness [duplicate]
Possible Duplicate:
How to understand the concept of compact space
Hello,
I am learning some analysis on my own and
what is the motivation to consider compactness?
eg. I do not understand why ...
- 317
2
votes
3
answers
369
views
How do we know that a map $f: U \to Y$ extends to $\bar{U}$?
I read the following fact: if $U$ is an open subset of $P_k^1$ and $f: U \to U$ is an automorphism of schemes, then $f$ extends to an automorphism of $P_k^1$. Thus I was curious: is there a general ...
- 25.6k
5
votes
0
answers
164
views
Group topologies on $\Bbb Z$ with dense open sets in $\Bbb T$
Let $\Bbb Z$ be embedded in the circle group $\Bbb T$ by an irrational rotation and regard $\Bbb Z$ as a subgroup/subspace of the topological group $\Bbb T$.
Are there group topologies $\mathcal A$ ...
- 1,145
5
votes
1
answer
523
views
Injections to binary sequences that preserve order
Suppose we have a countable set S with a total order. Can we give an injection from S to the set of finite binary sequences that end in all zeros that preserves the ordering? The order on binary ...
- 493
5
votes
2
answers
364
views
Complexity of a fixed point
Let $\varphi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ be a homeomorphism of
the plane with fixed point $p$, i.e. $\varphi(p)=p$, and no other periodic
points. Let $r$ be a fixed natural number. My ...
- 303
1
vote
0
answers
128
views
Properties of "incomplete finite simplicial complexes"
Definition: We say that $K'$ is an incomplete finite simplicial complex if there exists a finite simplicial complex $K$ such that $|K'|=|K|\backslash Y$ where $Y$ is a union of some open faces of K.
...
- 7,609
6
votes
0
answers
969
views
What relates to measure spaces as topological spaces relate to metric spaces ?
Has there been study of a generalization of measure spaces along the following or similar lines ?
Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball of radius $r$ of $U$ ...
- 529
2
votes
1
answer
245
views
Probability measures on $L^p$
Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
- 8,512
3
votes
1
answer
370
views
Weaker form of irreducible surjections
An irreducible surjection is usually defined as a continuous closed surjective map $f:X\rightarrow Y$ such that if for some closed set $C\subset X$ one has $f(C)=Y$ then $C=X$. In my dissertation I ...
- 2,275
4
votes
0
answers
172
views
$S^{3}$-valued harmonic analysis
Edit:
Note that $S^{3}$ with the quaternion operation is a group. For a locally compact Abelian group $\Gamma$ we consider
$$\tilde{\tilde{\Gamma}}=\{\phi:\Gamma \to S^{3} \mid \phi(xy)=\phi(...
- 356
1
vote
2
answers
166
views
TSP, but for all routes not all points
Hello
I am trying to solve a problem involving a cross-country ski trail map. I wish to travel every trail on the map, at least once, but no more than twice (so I can out-and-back on a destination, ...
- 113
10
votes
2
answers
367
views
existence of a connected set with given connected projections.
Suppose A and B are compact connected sets in the XY plane and XZ plane respectively in R^3. Suppose further that the the range of x-values taken by A and B are the same (i.e, projections of A and B ...
- 515
4
votes
1
answer
1k
views
A boundary-preserving map on the unit disk
We are given a (closed) ball $D^n$ and a (continuous) map $f: D^n \to D^n$, that is identity on the boundary of $D^n$.
Let $C$ be a subset of $D^n$, and let $f^{-1}(C)$ be the inverse image of $C$ ...
- 41
2
votes
1
answer
173
views
Is there a maximal (or maximal Tychonoff) non normal space?
Is there a maximal (or maximal Tychonoff) non normal space? In "A Problem of Set-Teoretic Topology" the existence of a maximal Tychonoff space is asserted. Also there exists a perfectly normal maximal ...
- 113
4
votes
2
answers
357
views
Minimal right ideals in finite semigroup
Let $E$ be a finite semigroup. According to N. Bourbaki (Algèbre I p. 121 exerc. 14 c), if $M$ and $M'$ are minimal right ideals in $E$, then they are isomorphic. I spent some time browsing through ...
- 43
8
votes
1
answer
688
views
Universal covers of domains in complex projective space
The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
- 4,485
1
vote
4
answers
5k
views
3
votes
1
answer
94
views
Special retraction from a metric space onto an arc
Suppose $X$ is a metric space and $A$ is a subspace of $X$ homeomorphic to $[0,1]$ with its usual topology. Let $v$ an end point of $A$, that is $v$ does not disconnect $A$. Is there a retraction $r$ ...
- 33
3
votes
0
answers
431
views
Bohr topos and quantization
Bohrification is a natural way to construct a quantum "phase space" (with some nice insights on foundational problems like non-contextuality through Kochen-Specker etc). I was wondering, since we get ...
- 585
3
votes
1
answer
958
views
When does an antipodal map on a manifold extend to the antipodal map on a spheres
So I have been mulling the following question over in my head for awhile now, and want to see if anyone else might have any ideas.
Begin with $M$ a manifold and suppose that $M$ has an antipodal map $\...
- 5,191
11
votes
0
answers
758
views
A basic question on Stone-Cech compactification of $\mathbb{Z}$
Can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? ...
- 895
2
votes
0
answers
87
views
Terminology for torsion semigroups where the order of elements is uniformly finite
A (multiplicatively written) semigroup $\mathbb A = (A, \cdot)$ with the property that ${\rm ord}_\mathbb{A}(a) := |\{a^n: n \in \mathbf N^+\}| < \infty$ for every $a \in A$ is called a periodic (...
- 10.5k
0
votes
1
answer
252
views
Which algebra of functions can be represented as $C(X)$
I don't know if this problem is known or not, so any information would be appreciated:
Let $\cal A$ be an $\Bbb{R}$-algebra of (not necessary continuous) real valued functions defined on a ...
- 9
3
votes
1
answer
617
views
How to make an ultranet
The only examples of ultranets/ultrafilters described in Bourbaki and Willard are the trivial ones (generated by a single point). I know that their existence relies in general on the axiom of choice ...
10
votes
0
answers
363
views
Krull dimension and Morley rank
Definition : A Topological space $\mathcal{D}$ is called noetherian if it satisfies the descending chain condition for closed subsets. We define the dimension of $\mathcal{D}$ to be the supremum of ...
- 1,872
4
votes
1
answer
562
views
Topology of the "normal spectrum" of a commutative von Neumann algebra
Kadison and Ringrose define normal states $\omega$ of a von Neumann algebra $A$ as such that $\omega(H_\alpha)\to \omega(H)$ for each monotone increasing net of operators $H_\alpha$ with least upper ...
- 7,426
2
votes
1
answer
128
views
Characterization of a subset of [0,1] $III$
I have a question related to the previous one.
Characterization of a subset of [0,1] $II$
Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e.
$t_n$ is said to converge to $...
- 1,835
0
votes
0
answers
153
views
extension of function in an abstract metric space
my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...
- 1,835
3
votes
2
answers
483
views
When does a LCA group not contain a (closed) infinite cyclic subgroup?
If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does not contain a closed (and hence discrete in the ...
- 3,115
2
votes
2
answers
293
views
Equivalence relations in suplattices
I am wondering about generalisations of the concept of equivalence relations to suplattices.
Here is my motivation: Given a set $X$. The powerset $\mathcal{P}(X)$ is a suplattice. For suplattices ...
- 2,442
7
votes
2
answers
419
views
Relation between $KO$ and $K$
What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. ...
8
votes
1
answer
223
views
local structure of free $\mathbb{R}$ actions
Assume the topological group $\mathbb{R}$ acts properly on a space $X$. Does then the projection map $p:X\rightarrow \mathbb{R}\backslash X$ have local sections ?
(for every $\mathbb{R}x\in \mathbb{R}...
- 11.1k
1
vote
1
answer
1k
views
Representations of regular maps (four color theorem)
For the scope of the four color problem and without lack of generality, maps can be represented in different ways. This is generally done to have a different perspective on the problem.
For example, ...
- 351
5
votes
1
answer
232
views
When can boundedness be characterized topologically in Metric spaces?
Let H be a separable and infinite-dimensional Hilbert space. Is every closed subset of H homeomorphic
to some closed and bounded subset of H?
- 6,215
2
votes
1
answer
145
views
Going Back-and-Forth Between Different Expressions/"Representations" for Open Books.
I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" ...
- 23
3
votes
1
answer
647
views
When is a sublevel set path-connected?
I am trying to completely characterize the conditions on $f : \mathbb{R}^n \to \mathbb{R}$ under which $\{x | f(x) \le 0 \}$ is path-connected. There are many obvious conditions that are sufficient (...
- 693
2
votes
1
answer
2k
views
Why not usual topology in measure theory ?
Measure theory was introduced in the early 1900s by Lebesgue, at the same time with Hausdorff introducing the usual concept of topology, and publishing it in his book just before World War I. Measure ...
- 105
0
votes
1
answer
107
views
Topology : Study on Separation Properties [closed]
I want an example of a completely regular space which is not a normal space. I have tried a lot but am unable to construct any example
- 115
0
votes
1
answer
2k
views
What does it mean to have Zero Density (mathimatically) [closed]
I read a question that asked "prove that the set of all positive integers expressible as the sum of two integers square has zero density." Now I was under the impression that something was dense iff ...
- 93
2
votes
1
answer
290
views
Idempotents in Green J classes
I recently read this article Syntactic semigroups. In page $8$, he speaks about a J class having an idempotent is called regular:
A $\mathcal J$-class containing an idempotent is called regular. ...
- 233
0
votes
1
answer
402
views
A question on cofinite topology.
Let $X$ be a countably infinite (or larger) set with the cofinite topology. for every $x\in X$ is there exists a family $\xi\subset\tau$ such that $\lbrace x\rbrace=\bigcap\xi
$ ? If the answer is yes,...
- 13
9
votes
0
answers
369
views
Is there Ultracoproduct-like construction for topological spaces in general?
In
http://arxiv.org/pdf/math/9704205.pdf
they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...
- 241
3
votes
2
answers
225
views
For any entourage $U,V$ there's an entourage $W$ such that $U\circ W\subseteq V\circ U$
Let $(X,\mathcal U)$ be a uniform space and let $U\in \mathcal U$. Is this statement true?
$$\forall V\in \mathcal U, \exists W\in \mathcal U, U\circ W\subseteq V\circ U$$
I think if the above ...
- 63
8
votes
0
answers
508
views
Is there in ZFC a topological space which is normal, ccc, countably compact, first countable and non-compact?
I am looking for a space as in the title and since many very similar spaces do exist in the literature, I wonder whether someone has a reference (different from the ones I cite below) or just some ...
- 1,855