All Questions
5,184 questions
5
votes
1
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498
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Percolation in Cayley graphs of semigroups.
Percolation in Cayley graphs of groups are studied by many researchers. There are also the concept Cayley graphs for semigroups. Are there any research about percolation in Cayley graphs for ...
- 6,211
8
votes
0
answers
299
views
Spaces that never separate the Hilbert cube
I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.
Any finite dimensional space has this ...
- 29.1k
1
vote
0
answers
173
views
Does real analytic imply locally contractible?
The statement is true for complex analytic spaces. I am not sure who proved this result.
I ask the same question in the real case.
- 85
4
votes
1
answer
1k
views
properly interpreting Pi_0 in the homotopy exact sequence
Define the lens space L(m,n) as the quotient of S2m+1 by the action of the cyclic group ℤn⊂S1⊂ℂ*. We can create the infinite lens space L(∞,n) by a telescoping construction ...
- 6,069
15
votes
0
answers
1k
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Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?
This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...
- 37.8k
2
votes
1
answer
89
views
Constructivity of zeros demanded by topological degree
Let $f : S^{n - 1} \to S^{n - 1}$ be a smooth map from the unit vectors of $\mathbb{R}^n$ to themselves. If $f$ has nonzero degree, then we know that any smooth map $g : D^n \to \mathbb{R}^n$ ...
- 5,827
6
votes
1
answer
727
views
Homomorphisms of Topological Groups which are Automatically Fiber Bundles?
Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will ...
- 27.5k
6
votes
0
answers
223
views
Series in topological rings that only converge if almost all summands are zero
While trying to understand a certain topological ring better, I stumbled onto the following question.
Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a ...
- 9,650
2
votes
1
answer
431
views
Automorphism of first homology and mapping class group
It is known that for a torus $\Sigma$, every automorphism of $H_1(\Sigma; \mathbb{Z})$ is induce by an orientation preserving self-homeomorphism of $\Sigma$ unique up to isotopy. In onther words, ...
- 93
8
votes
2
answers
592
views
Base change for category objects in topological spaces
I was prompted by this question, but the motivation is different.
Suppose we have an internal category object in topological spaces, i.e. an object space X and a morphism space Y, together with ...
- 52.6k
6
votes
0
answers
618
views
Duality between conjugacy classes and irreducible characters for finite monoids?
Qiaochu's answer to this question suggests that the proper way to view the bijection between conjugacy classes and irreducible complex representations of a finite group is via a duality. My question ...
- 38.6k
3
votes
1
answer
737
views
Klein Bottle exception to the Heawood Conjecture [duplicate]
Possible Duplicate:
The Klein bottle and the Heawood Conjecture
It is well known that the Heawood Conjecture states that the bound for the number of colours which are sufficient to colour a map ...
- 1,441
9
votes
1
answer
626
views
Stable presentable categories as module categories
There is a theorem of Schwede and Shipley which classifies categories of modules over an A∞ ring spectrum as those stable presentable (∞,1)-categories with a compact generator. Suppose I ...
- 25.2k
4
votes
1
answer
798
views
Topological dimension, is it local?
Let $n\in\mathbb N$ and $X$ be a complete metric space.
Assume that there is $\epsilon>0$ such that
$$\dim B_\epsilon(x)\le n$$
for any $x\in X$.
Is it true that $\dim X\le n$?
Here $\...
- 1,785
7
votes
0
answers
466
views
Closure properties of familes of $G_\delta$ sets.
Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space ...
- 17.6k
3
votes
1
answer
2k
views
What is the pure intuition for topological continuity and topology? [closed]
I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity.
The ...
- 191
6
votes
0
answers
105
views
Large discrete subspaces in spaces of separately continuous functions
For topological spaces $X,Y,Z$ let $SC_p(X\times Y,Z)$ be the space of separately continuous functions $f:X\times Y\to Z$ endowed with the topology of pointwise convergence.
It is easy to see that ...
- 42k
8
votes
0
answers
403
views
Is the product of a discretely Lindelöf space with [0,1] discretely Lindelöf ?
A space $X$ is discretely Lindelöf iff given any discrete subset $D$ of $X$, its closure in $X$ is Lindelöf. Such spaces were introduced by Arkhangel'skii about 15 years ago (if I am not mistaken) ...
- 1,855
8
votes
0
answers
247
views
Construct a topologically $\infty$-dimensional separable metric space.
But don't assume knowledge of any topological dimension theory. Here is a specific approach (an open problem):
Does there exist a separable metric space $X$ such that the following two conditions ...
- 7,500
0
votes
1
answer
57
views
If $X$ has the "discrete" covering property, how about $X^2$?
We say that a space $X$ has covering property (C) if the following holds:
(C) For any open cover ${\cal U}$ of $X$ there is a closed discrete set $D\subseteq X$ and a map $\varphi: D\to {\cal U}$ ...
user70876
3
votes
2
answers
467
views
Euler characteristics and operator indices as exponents for Laurent polynomials
This question is rather vague. Are there any natural situations which involve Laurent polynomials of the form
$$\sum q^{a_i}\in\mathbb{Z}[q,q^{-1}]$$
where the $a_i$'s are either Euler characteristics ...
- 1,129
1
vote
0
answers
73
views
Injectively rigid spaces
Given a set $X$, is there a topology $\tau$ such that the identity $\text{id}_X$ on $X$ is the only continuous injective self-map?
(This is Joel David Hamkins's recent question in the category $\...
- 48.9k
2
votes
0
answers
87
views
Local section of Lie Groupoids
Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...
- 21
1
vote
1
answer
1k
views
Maximal ideals and ultrafilters [closed]
I am not sure about these two definitions.
For example, if we take the power set of A={1,2,3} with the partial order of inclusion. What are the maximal ideals and what are the maximal filters? For ...
- 11
7
votes
0
answers
626
views
Does local strict contractibility imply ANR?
Say that a space (= compact metrizable space) $X$ is locally strictly contractible if, for every $p\in X$ and neighborhood $U$ of $p$, there is a neighborhood $V$ of $p$ which can be contracted to $p$ ...
- 2,049
0
votes
1
answer
128
views
What are the semigroups in which congruence classes can be multplied like sets?
For a semigroup $S$ and a congruence $\rho$ on $S$, let's say that $\rho$ is good when for all $a,b\in S$ we have that $[ab]=[a][b],$ where $[x]$ denotes the congruence class of $x$ modulo $\rho$ and ...
- 1,445
0
votes
1
answer
278
views
On the compactness of a certain chain topology [closed]
Let $X$ be a non-empty set and $I$ a collection of some nested subsets of $X$ indexed by a linearly ordered set $(\Lambda,\le)$ such that $I$ always contains the void set $\emptyset$ and the whole set ...
- 243
4
votes
2
answers
686
views
Determining if two algebraic sets are homeomorphic
Is there an algorithm which, given two polynomials in $n$ variables with real coefficients, $p(x)$, and $q(x)$, will determine whether the zero sets $p^{-1}(0), q^{-1}(0)\subset R^n$, are homeomorphic ...
- 520
2
votes
1
answer
911
views
A density condition for metric spaces
I have encountered the following property. Can anybody tell me if it already exists in literature and/or is equivalent/similar to other well-known properties?
Property: $(X,d)$ metric space. For ...
- 6,034
3
votes
1
answer
594
views
Powers of quotient maps
It is well-known that if $q:X\to Y$ is a quotient map, then the self-product $q^2:X^2\to Y^2$ need not be a quotient map. For instance, if $X$ is the real line generated by the basic sets $(a,b)$ and $...
- 7,193
2
votes
1
answer
360
views
On a property of subsemigroups
Let $H$ denote a subsemigroup of a semigroup $G$.
I'm interested in the following property:
$$\forall g\in G\exists h\in H:gh\in H.$$
This property is weaker than the property that $H$ is an ideal ...
- 21
1
vote
0
answers
62
views
Reference request - Compact embedding of intermediate space
Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...
- 679
4
votes
2
answers
226
views
Modal models as reduced products?
In model theory for standard first-order logic, one constructs a single model, a reduced product, from a collection of first-order models, together with an index set and a filter on the index set.
In ...
- 327
2
votes
1
answer
422
views
Is every zero-dimensional space with no infinite clopen partition pseudocompact?
For this question, we say that a zero-dimensional space $X$ is $\omega$-pseudocompact if every partition of $X$ into clopen sets is finite. In other words, a zero-dimensional space $X$ is $\omega$-...
2
votes
0
answers
212
views
Can a compact metrizable space be determined by its Hausdorff measures?
Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define $$L(d,h)=\lim_{\...
1
vote
1
answer
334
views
topological equivalence of ODEs
Let $n$ be a non-negative integer. $\;\;$ Let $\: f : \mathbb{R}^n \to \mathbb{R}^n \:$ and $\: g : \mathbb{R}^n \to \mathbb{R}^n \:$ be Lipschitz.
Define the relation $\stackrel{f}{\sim}$ on $\...
user5810
4
votes
1
answer
312
views
General topology terminology questions
In a Hausdorff but not regular space, collapsing certain closed sets to a point may produce a non-Hausdorff space. Does there exist a term for closed sets one may collapse and still have a Hausdorff ...
- 17.6k
3
votes
1
answer
502
views
Determining continuous functions on Banach spaces
Let $X$ be a real Banach space.
For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
- 11.5k
13
votes
0
answers
1k
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Paracompact Hausdorff but not compactly generated?
I'm sorry to be asking a (possibly) elementary question, but I've run into a problem in point-set topology; I've just read that there exists paracompact Hausdoff spaces which are not compactly ...
- 15.5k
2
votes
0
answers
1k
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Double Torus Parametric Surface [closed]
In the process of trying to find continuous parametric surface equations for the double torus and for a pair of pants, I believe that the problem is unsolvable for some topological reason.
I have ...
0
votes
2
answers
478
views
"Exotic" Banach spaces of sequences
Does there exist a linear subspace of $\mathbb C ^{\mathbb N}$ that can be endowed a Banach space topology that is not finer than the locally convex topology of pointwise convergence?
Best,
Martin
- 5,327
-1
votes
1
answer
81
views
extension of a continuous function [closed]
Please is it true that if $f:K\to \mathbb{R}$ is a continuous function of a comact set $K\subset\mathbb{R}^m$ then $f$ can be extended to a continuous function of some open neighbourhood of $K$?
...
- 1
1
vote
1
answer
260
views
The intersection of Block Groups and R-trivial (finite) monoids
Let $\textbf{BG}$ be the pseudovariety of block groups, also known as $\textbf{EJ}, \textbf{PG},\ldots,\text{etc.}$(see [1]), and let $\textbf{R}$ be the pseudovariety of R-trivial monoids, by the ...
- 424
3
votes
1
answer
353
views
Topological space with some conditions
Can one give an example of non-compact space $X$ which satisfies the following conditions:
the countable union of compact subsets is relatively compact,
for every closed noncompact subset $A$ of $X$ ...
- 145
1
vote
0
answers
430
views
Intersection of cocompact closed normal subgroups
Let $G$ be a locally compact Hausdorff topological group.
Definition A closed normal subgroup $H \unlhd G$ is called cocompact if $G/H$ is compact with respect to the quotient topology.
Note that ...
- 1,498
0
votes
1
answer
52
views
Compact $R_1$-spaces
A space $(X,\tau)$ is said to be $R_1$ if for all $x,y\in X$ with $cl(\{x\}) \neq cl(\{y\})$, there are disjoint open sents separating $cl(\{x\})$ and $cl(\{y\})$.
If $X$ is compact and $R_1$, does ...
user62017
0
votes
0
answers
136
views
Monoid action on an uncountably infinite set
The action of a monoid on a finite set is equivalent to a finite state machine, however I would like a categorical way to think about an uncountably infinite state machine (a state transition system?)....
- 101
2
votes
0
answers
181
views
A categorical analogue of Debreu's independent factors theorem
Background
A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...
5
votes
0
answers
101
views
The name for the quotient property
I asked this question on math@stackoverflow and was suggested to ask it here as well.
We call a surjective $f:X\rightarrow Y$ a quotient mapping if it satisfies, for every $U\subset Y$
(continuity,...
- 187
1
vote
2
answers
360
views
Is this quotient space of Q_p contractible?
Let $X_{p} = \mathbb{Q}_{p} / \sim $, where $\sim$ is defined by:
$x\sim 0 \Leftrightarrow x\in \mathbb{Q}$
$X_{p}$ is path-connected, because (unless I'm making some horrible mistake,) for any $x\...
- 713