All Questions
5,184 questions
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Defining connectivity between K points on a periodic domain in terms of proximity
THE SITUATION:
Begin by taking a periodic strip of length 2*Pi. Then use a uniform distribution to place K points (x1,…, xk) on the strip by assigning each of them a randomly sampled number. Then ...
- 11
4
votes
0
answers
210
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properties of $\beta\omega\setminus\omega$ minus the P-points
Let $X=\beta\omega\setminus\omega$ and let $Y=X\setminus P$ where $P$ is the set of P-points in $X$. Then $P$ is dense in $X$ if we assume CH but $P$ may be empty otherwise (Shelah). In the one case $...
- 607
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223
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A closure operation on subsets of ${\Bbb Z}[x]$
Given a(n infinite) set $S\subset {\Bbb Z}[x]$ (integer polynomials), write $R_S$ for the topological closure of the set of all complex roots of all $p\in S$. Then write $\hat{S}$ for the set of all ...
- 17.6k
4
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1
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417
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"Category" of Nonempty Metric Spaces and Contractive Maps?
The usual way of getting a category of metric spaces is to take metric spaces as objects, and the nonexpansive maps (ie, functions $f : A \to B$ such that $d_B(f(a), f(a')) \leq d_A(a, a')$) as ...
- 9,201
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169
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Algebraic properties of the semiring of open subsets.
Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at all?...
- 3,513
5
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answers
501
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Profinite topologies
We can define two topologies on a group $G$ by considering all normal subgroups of finite index (resp. of index a finite power of $p$ - where $p$ is a prime) as basis of $1\in G$.
My questions: Under ...
- 51
3
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321
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Removing intersections of curves in surfaces
Let $C_1, \dots, C_n$ be a family of disjoint simple curves in a surface $\Sigma$. If $C$ is any simple curve in $\Sigma$, it turns out that we can map $C$ to a curve $C'$ (via a homeomorphism of $\...
- 32.1k
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1
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387
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Why not use usual topology in ordered spaces ?
This is a similar question to the one about the lack of use of usual topologies in measure theory. By usual topology here is meant the Hausdorff-Kuratowski-Bourbaki concept, based on open sets, or ...
3
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1
answer
828
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When is the realization of a simplicial space compact ?
Suppose $X$ is a simplicial space of dimension $M$ (i.e. all simplices above dimension $M$ are degenerate). The claim is:
$|X|$ is compact. iff $X_n$ is compact for each $n$.
Suppose each $X_n$ is ...
- 11.1k
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1
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280
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"Skein" equations sets that can reduce any graph
Consider for example the approach to the Kuperberg G2 or the Yamada polynomial. I don't know whether relations of graph (in contrast to knot) theory are also usually called "skein" but
I simply carry ...
- 4,829
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1
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217
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F-spaces and points whose complements are C*-embedded
Let $X$ be a compact Hausdorff space. If $X$ is extremally disconnected then the complement of every point $x$ in $X$ is C*-embedded in $X$ (i.e every continuous bounded real-valued function on $X\...
- 607
1
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1
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362
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Winding number bijection on graphs
Let $G=(V,E)$ be an isoradial graph. In other words the graph can be imbedded into the plane such that each face (plaquette) can be circumscribed a circle of radius 1 with the circle's center ...
- 4,952
2
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1
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254
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An ordinal invariant for spaces based on a hierarchy of closed sets from z-sets
EDIT: I apologize for the confusion by which I originally framed this question for normal spaces, where it has an uninteresting answer (thanks to those who pointed this out). Hope I've got it right ...
- 17.6k
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72
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Decomposition results for locally commutative semigroups
Every finite abelian group is the direct product of its cyclic groups of prime order, and every commutative monoid divides a product of its cyclic submonoids. Could these results generalized to ...
- 798
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455
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Quotients of topological groupoids
The issues that arise when moving from topological groups to topological groupoids are (at least to me) both subtle and interesting. Recently, I was reading a paper of R. Brown and J.P.L. Hardy from ...
- 7,193
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319
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Hilbert space automorphisms realized as induced by transformations of some base-spaces
Following question may be soft. Fix abstract hilbert space H and consider any automorphism A in banach-spaces sence (i.e. no conditions on metric). Call A is realizable if exist measure space $(X,\mu)$...
- 789
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721
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What is the horn torus homeomorphic to?
Is the horn torus homeomorphic to some other well known object? In particular, the standard torus can be described by a square with collapsed edges. What about the horn torus?
- 1,638
1
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75
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Collapsing a countable collection of intervals on $\mathbb{S}^1$
Consider a countable collection $I_n$ of closed connected disjoint intervals on $\mathbb{S}^1$. When this collection is maximal, the set $\bigcap \nolimits_{i=1}^{n}( \mathbb{S}^1 \backslash \bigcup \...
- 541
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1
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322
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Topological Properties of Non-Smooth Functions [closed]
I am interested to know what topological properties are possessed by non-smooth functions. I suspect this is well known, but I can't find what I am looking for from google.
Is there a topological "...
- 137
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1
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390
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Isocontours of depth and magnitude of gradient
We are interested in characterizing a 2D surface $z(x,y)$, where $(x,y)$ is the regular 2D Cartesian grid. Let $\nabla z = (z_x, z_y)$ denote the gradient. The surface is a "general" one, that is, ...
- 13
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1
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336
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cardinality of final coalgebras in Top
Let P be a polynomial functor from Top to Top, by which I mean a functor of the form P(X) = ∐i ≥ 0 Si × Xi where the Si are finite sets, all but finitely many of which are empty. ...
- 25.2k
6
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510
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The Mapping Cylinder of a Pullback Square
Suppose I have a pullback square, which I think of as a map from the fibration
$q:X\to A$ to the fibration $p:Y\to B$. Then there is an induced map $m: M \to N$
from the mapping cylinder $M$ of $X\...
- 12.5k
2
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1
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243
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Induced pretopologies on sSet
Recall that the geometric realisation functor $| - |: sSet \to Top$ preserves products (choosing $Top = k Space$ or similar). Thus any given singleton Grothendieck pretopology on $Top$ gives rise to a ...
- 35.5k
1
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202
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Soft sheaves on indiscrete paracompact spaces
Let $X$ be some space, I have basically 2 questions:
1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on ...
- 417
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0
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365
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Finding paths in a path connected space
I'm looking for such literature as exists relevant to the following problem.
Problem Given a compact, path-connected region $E$ on the plane and a positive constant $r$. Find (if possible) a path ...
- 627
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1
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242
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Are mapping spaces paracompact?
Let X be a (finite dimensional) manifold. Consider smooth mapping space $$PX = C^\infty(I, X)$$ where I = [0,1] is the closed interval. Is this space paracompact? What if we fix a point x in X and ...
- 27.5k
1
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1
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262
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$\omega$-monoids
Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question:...
- 1,882
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94
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Extending coverings over dense subsets
Let $X$ be a metric space with $D⊆X$ a dense subset.
If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?
For a ...
- 1
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1
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270
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Random question: Is there a set-theoretic description of projective space? [closed]
I met projective space via a recent class on perspective drawing, believe it or not, but I didn't know that this was the "space" we were using. I came across a more detailed description trawling the ...
- 91
6
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1
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187
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Homotopy type of stabilizers
Let X be a contractible metric space and G a topological group acting transitively on X (i.e. given any two points x,y \in X, there exists g \in G such that gx=y).
My question is the following: is it ...
- 1,060
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433
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Ever seen a ringed group?
A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...
- 12.3k
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1
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87
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Question regarding closure of sets defined by the vanishing of holomorphic functions
Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, g$...
- 3,245
4
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354
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Terminology for topological base closed under intersection?
Is there an established or well justified terminology for a topological base that is closed under finitary intersections?
As motivation, recall these conditions on a collection of subsets of a given ...
- 2,754
1
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1
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72
views
Transformation terminology question
Given a transformation $t$ from the transformation semigroup $T_{n}$, if you take powers of $t$ under composition you get a length $s$ stem followed by a cycle. Permutations by definition have a ...
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0
answers
558
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continuous selection of a multivalued function?
The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...
- 1,839
1
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0
answers
267
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subset embedding gives trefoil knot [closed]
Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$.
It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that
the embedding $S^1\...
- 11
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310
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The self-duality of topological compactness
The impatient reader can skip my attempt at motivation and go straight my "Question formulations for the impatient."
In a failed(?) attempt at discovering something new, some years ago I ...
- 17.6k
2
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0
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146
views
How do you call a map which sends convergent sequences to pre-compact ones ?
In my work I encountered a map $f$ between two metric spaces $X$ and $Y$ that was not continuous (at least I couldn't prove it was), but I was able to prove that convergent sequences $(x_n)$ in $X$ ...
- 4,123
6
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360
views
The Space of Cellular Maps
Let $X$ and $Y$ be CW complexes.
Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...
- 12.5k
1
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1
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208
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When do maps of ineffective orbifolds descend to their effective part?
If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between ...
- 15.5k
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1
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271
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Numbers associated with boundaries of manifolds
I don't know what name if any is attached to the numbers I'm about to describe.
For a line segment, [a,b]
the number is 1 if for any k in (a,b)
and 2 if k=a or k=b.
For a square, [a,b] ...
- 379
2
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1
answer
336
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Topologies making a class of functions continuous [closed]
Let $X:=\{f: \mathbb{C}\to \mathbb{C}\}$ be a class of total functions on $\mathbb{C}$ closed under composition, addition, multiplication, and scalar multiplication. Does there exist a topology on $\...
3
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0
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251
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What is the origin of the metrization problem for compact convex sets?
The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...
- 3,547
1
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1
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131
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Conditions under which a given scheme converges
I'm sorry in advance for how long this question is. Suppose I have a continuous function $f:\mathbb{R}^n \rightarrow \Delta_{n-1}$, where we think of the simplex $\Delta_{n-1}$ as the set
$\Delta_{n-...
- 645
2
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1
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153
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chains and countability
Given a point $x$ in a topological space $X$. I was wondering, whether one can always find a local basis at $x$, which is totally ordered (a chain) under inclusion. For example this is true for spaces,...
- 11.1k
2
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0
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167
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Local cartesian closedness in the category of compactly generated spaces
According the the nLab, the category of compactly generated (CG) spaces is not locally cartesian closed.
So if $A$ is a CG space and $C$ a CG space above $A$, $C$ may not be exponentiable.
What if we ...
- 3,033
2
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1
answer
265
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Hausdorff Derived Series
There is a short section in the book Locally Compact Groups by Markus Stroppel (Chapter B7) on the notion of a "Hausdorff Solvable Group", which he defines as a topological group with a descending ...
- 33
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0
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285
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How to use the Lefschetz trace formula on infinite dimensional spaces?
I think the Lefschetz trace formula says something like:
if $F: X \to X$ is a continuous map of compact manifolds, then
$\chi(X^F) = \sum (-1)^i \mathrm{Tr} f_*|_{H_i(X)}$
First of all, this ...
- 8,723
2
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0
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254
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Simple terminology question about the Dubrovnik (Kauffman) polynomial
In my S matrix classification attempts I encounter a lot of
Dubrovnik polynomials of the form D(z-1/z,z^n) and D(-z+1/z,z^n).
[Second variable is for writhe, n is an integer; for the first I don't
...
- 4,829
2
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0
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140
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Products for probability theory using zero sets instead of open sets
(For all of this post, at least Countable Choice is assumed to hold.)
For all Tychonoff spaces $\langle X,\mathcal{T}\hspace{.06 in}\rangle$ :
Define $\mathbf{Z}(\langle X,\mathcal{T}\hspace{.06 in}\...
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