All Questions
5,184 questions
2
votes
1
answer
404
views
Follow up question on union of disjoint Vitali sets...
Since I haven't received a satisfactory answer to my initial question I'm going to ask a somewhat weaker one...
This time we say $X$ is a Vitali set in the closed interval $[0, 1]$ with respect to $\...
- 463
2
votes
1
answer
116
views
Composition of (topologically) connected binary relations
My question seems far too basic to be unknown, but I could not find anything relevant...
Let $X$, $Y$ and $Z$ be compact connected metric spaces, and let $F \subset X \times Y$ and $G \subset Y \...
- 4,010
4
votes
2
answers
551
views
Normality of an affine semigroup
An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is ...
- 111
4
votes
2
answers
439
views
Legendrian homotopy of curves in a contact structure?
I'm aware of the great body of work on Legendrian knot theory in contact geometry, but suppose I'm curious just about homotopy and not isotopy. How does one understand the space of Legendrian loops ...
- 13.6k
1
vote
1
answer
179
views
Measures idempotent with respect to addition and multiplication.
Does there exist a probability finitely additive measure on $\mathbb N$ which is idempotent with respect to addition and multiplication simultaneously?
It is known (due to Hindman) that there is no ...
- 723
3
votes
2
answers
300
views
Discriminant locus in knot space
Consider the space $K$ of all immersions of $S^1$ into $\mathbb R^3$.
The set of knots with self-intersection is a discriminant in $K$ and divide it into "chambers".
Let $f$ be a knot with $n$ double ...
- 5,063
2
votes
1
answer
1k
views
Finding saturated open sets
Suppose I have a continuous map $f:X\rightarrow Y$. Then one can wonder, whether for every open set $U\subset X$ the set
$U':=\{x\in X|f^{-1}(f(x))\subset U\}$ is open again. This is not true in ...
- 11.1k
7
votes
0
answers
299
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
4
votes
0
answers
396
views
Is there a homological way to compute quiver presentations?
I have recently been studying with colleagues the representation theory of certain finite monoids that come up in probability theory and combinatorics, see Ken Brown's beautiful survey here.
These ...
- 38.6k
0
votes
0
answers
84
views
Can a "weak" topological space be a Moore space?
Let B be a reflexive and infinite dimensional real Banach space-which could be Hilbert space l^2- and let B be endowed with the weak topology. Although this topology is regular and Hausdorff, it is ...
- 6,215
-1
votes
1
answer
75
views
Finiteness of "novel variance" from a kernel on a compact space [closed]
Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...
- 8,512
3
votes
1
answer
361
views
Is the coproduct of fibrant spectra fibrant again?
Define an $S^{1}$-spectrum $E$ to be a sequence of pointed simplicial sets $E_{n},\\ n=0,1,2...$ with assembly morphisms $\sigma_{n}:S^{1}\wedge E_{n}\rightarrow E_{n+1}$.
An $S^{1}$-spectrum $E$ is ...
- 51
1
vote
1
answer
909
views
What are the topological properties of the metric space retained (inherited) for its completion
Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained (...
4
votes
1
answer
479
views
"monotone" homotopy?
This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations.
Let $X$ be a (bounded) metric space, $Y$ be a topological space and $A\...
- 101
1
vote
0
answers
430
views
Universal Hausdorff Space [duplicate]
Possible Duplicate:
Largest Hausdorff quotient
Is there a left adjoint to ${\mathbf{Haus}}\to{\mathbf{Top}}$? Here ${\mathbf{Haus}}$ is the full subcategory of Hausdorff spaces in ${\mathbf{Top}}$...
user2529
0
votes
1
answer
493
views
Sheaf of sections and local triviality
This is probably not a research level question, I'm sorry if it is inappropriate. I'm reasking here this question on math.se.
Suppose that $\xi: E \to B$ is a bundle (by which I mean simply a ...
2
votes
1
answer
259
views
Nuclearity of certain semigroup crossed product C*-algebras
This question is related to this question link.
Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. ...
- 2,029
6
votes
1
answer
442
views
Countable paracompactness, normality and locally countable open covers
(repost from the topology Q&A board)
I have a (T_1), Normal, countably paracompact space X. I would like to know if every locally countable open cover of X (i.e. an open cover such that every x ...
- 1,331
5
votes
1
answer
294
views
Isotropic subspaces in cohomology
Hello,
Here is a problem I encountered in the study of Kähler manifolds but there is a natural generalisation of this for topological spaces.
If $X$ is a topological space, denote by $g_\mathbb{R}$ ...
- 339
-2
votes
1
answer
780
views
commutative monoids have binary products? [closed]
Does the category CMonoid of commutative monoids have binary products?
thanks
- 1
4
votes
1
answer
511
views
A question about open subsets of Hilbert space
If H is (a separable and infinite dimensional) Hilbert space and if U is a non-empty
open subset of H that is not connected, does the boundary B of U always have at least
one component that is not a ...
- 6,215
1
vote
1
answer
324
views
Sufficient conditions for Hausdorffness
Let $(X,\tau)$ be a $T_1$ topological space and $Y\subset X$ a dense subspace which is completely metrizable. Are there any sufficient conditions to ensure that $(X,\tau)$ is Hausdorff using the known ...
- 159
0
votes
2
answers
210
views
Locally compact, 0-dimensional, pseudocompact space
Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional?
It is known that a 0-dimensional locally compact space which ...
- 1,704
3
votes
0
answers
637
views
Fixed point theorem for convex, closed multivalued mapping
There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:
Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
- 696
9
votes
0
answers
236
views
H-spaces without rational homology
Does there exist a simply connected, non-contractible manifold $M$, which is an $H$-space,
and whose rational homology groups vanish in positive degrees?
My space $M$ is in fact homotopy equivalent ...
3
votes
1
answer
587
views
Functoriality of base change
Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism $\kappa$...
- 1,457
3
votes
0
answers
176
views
Extending a Hilbert space isometrically
Let $H$ be a Hilbert space, and let $X$ be a topological vector space.
Under what conditions on the topologies of $X$ and $H$ does there exist an injective, continuous linear map $f : H \to X$?
...
- 8,512
4
votes
0
answers
939
views
Proofs of Baire category theorem
I would like to have a list of proofs of the fact that the real line is not meager (also very useful would be a reference to such a list, if it already exists somewhere).
My motivation is the ...
- 315
3
votes
1
answer
164
views
Algebras with countable chains only
Is there an example of an uncountable Boolean algebra $B$ in which every chain is countable and such that $\ell_\infty$ embeds into the Banach space $C(\mbox{Stone }B)$? The latter requirement is not ...
- 387
3
votes
1
answer
528
views
A question about connected inner limiting sets
Let M be a finite-dimensional Euclidean space or an infinite-dimensional separable Banach space.
An inner limiting subset of M is a countable intersection of open subsets of M-these sets are
usually ...
- 6,215
2
votes
0
answers
194
views
realcompact space
I want to study realcompact spaces but I can't find the best book or journal for it, and I really need to.
(sorry I don't write English very well)
- 21
4
votes
1
answer
354
views
Does the weak approximation theorem hold for general topological fields?
The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...
- 2,585
1
vote
0
answers
245
views
Sums of Strongly z-ideals
In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called strongly $z$-ideal if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} \mathcal{M_{\alpha}}$...
- 1,788
1
vote
1
answer
317
views
Mapping class group and cylindrical structure
Let us fix a torus $\Sigma=S^1 \times S^1$. We consider a cylinder $\Sigma \times I$ and a data $(\Sigma\times I, \Sigma\times 0, \Sigma\times 1, f_{0},f_{1})$. Here $f_{i}$, called parametrization, ...
- 93
1
vote
1
answer
400
views
Transitive Semigroups of $2\times 2$ matrices
Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there ...
- 1,045
0
votes
1
answer
296
views
homeomorphism of topological group
Let G be a topological group and a be an element of order 2 in G. Further suppose the element a does not belong to center of G. Then is it true that only homeomorphism f of G such that $f(ax)=f(x)a$ ...
- 1
2
votes
1
answer
381
views
A question about disconnecting a Euclidean space or a Hilbert space
By a "totally disconnected" point set I mean one whose only connected subsets are singletons. Can a finite dimensional Euclidean space whose dimension is at least two,
be separated by any subset that ...
- 6,215
3
votes
1
answer
405
views
Topological simplicity and dense subgroups
Let $G$ be a (topologically) simple Hausdorff topological group. Let $H$ be a dense subgroup of $G$. Now throw away the topology. What restrictions are known on the structure of $H$ as an abstract ...
- 4,728
2
votes
0
answers
72
views
d-refining covering of normal space
If $X$ is normal, it is well known that for any open-covering $(U_i)$ of $X$, there exist closed subspaces $F_i$ and $G_i$ and an open subspaces $O_i$ such that $$F_i\subset O_i\subset G_i\subset U_i\...
- 189
3
votes
1
answer
233
views
Continuity of the maxima
Let $f:X\times Y\mapsto R$ be jointly continuous, where $X$ is some topological space, $Y$ is some countably compact topological space. My question is whether or not the envelop $\phi(x) := \max_{y\in ...
- 101
2
votes
1
answer
321
views
CG Hausdorff space
Let X be in CGHaus and Y locally compact hausdorff. The usual product space XxY is CGHaus, so we dont need to apply that special functor to it (the one that takes a space to the space with same points ...
- 21
0
votes
0
answers
109
views
Characterising singular homology among a more general class of cosimplicial spaces
Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
- 1,105
0
votes
0
answers
218
views
When is $\{ x \ge 0 | f(x) \le 0\}$ path-connected?
I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x \ge 0 | f(x) \le 0 \}$ is path-connected. We can assume that $f$ is continuous and concave.
...
- 693
3
votes
1
answer
419
views
Question on coverings and and their classifying spaces [closed]
Hello. I have a question on covering spaces. Please let each space be paracompact, path-connected, locally path-connected and semi-locally simply connected.
Let $E\to B$ denote a normal covering ...
- 31
4
votes
1
answer
382
views
Is the subspace of DVR's of the Zariski-Riemann space still quasi-compact?
If $S$ is the Zariski-Riemann space of a noetherian subring $k$ of a field $K$, Zariski-Samuel prove that $S$ is quasi-compact. If $S'$ is the subspace of valuations that are discrete (i.e. that ...
- 1,347
2
votes
1
answer
159
views
Continuity at a point in sequential spaces
Let $X$ be a sequential space, $Y$ be some topological space, and $f:X\mapsto Y$ define some function. If $\forall x_n \to x, n\in\mathbb{N}$ implies $f(x_n) \to f(x)$, then does it follow that $f$ is ...
- 101
1
vote
0
answers
91
views
Tubular neighbourhood which is nowhere piecewise linear
I recently asked this question.
I think, if the following were true, then I would solve my problem.
Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...
- 11
4
votes
0
answers
1k
views
Associative binary operations on natural numbers
Which are all the associative binary operations on natural numbers ?
Certain results in this regard can be found in arxiv:math/0508215.
It appears that such associative operations cannot grow too fast....
1
vote
2
answers
394
views
Relations in matrix semigroups
Suppose that $A_1, \dots, A_k \in M_n(\mathbb{Q})$ and $S$ is the semigroup generated by them. Two questions: are there always a finite set of relations $\{R_i\}$ among the $A_j$ such that $S$ is ...
- 4,646
2
votes
0
answers
564
views
Direct Limits and Limits of Nets
A net is a function from a directed set into a topological space, and it is said to converge to a point if certain conditions are satisfied. Similarly, a direct system is a function from a directed ...
- 15.4k