All Questions
5,184 questions
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
-2
votes
1
answer
458
views
some trouble over the cardinality of the cantor set(middle one-thirds) [closed]
firstly i thank you for taking interest in my post but i am new here so if i have made some mistakes or done something which is out of place please point out.my problem is-
we know that the cantor ...
- 19
21
votes
1
answer
1k
views
Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?
Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra.
Moreover, a morphism of commutative von Neumann algebras induces
a continuous morphism of the corresponding ...
- 37.8k
8
votes
4
answers
3k
views
Finite dimensional vector spaces over a complete but not-necessarily-valued field
I'm essentially reopening this old question of Ricky Demer which was never fully answered.
Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...
- 2,585
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
5
votes
2
answers
641
views
Shortest "painting" of the sphere
Let $S$ be the sphere in $\mathbb{R}^3$ and $C:[0,1]\to S$ a continuously differentiable curve on $S$. Let $T:[0,1]\to\mathbb{R}^3$ denote the tangent vector of $C$. Let $P(t)$ be the plane containing ...
- 1,533
1
vote
0
answers
267
views
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
0
votes
0
answers
850
views
Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$
I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:
We have a collection of curves $c_i$ for $i=1,2,..,n$ embedded in ...
- 1
7
votes
1
answer
266
views
Positive cone of a subgroup of $\mathbb{Z}^n$
This question sounds like it should be very well known, but for some reason I failed to find a decent answer anywhere. Let $G\subset\mathbb{Z}^n$ be a subgroup, and $G_+=G\cap\mathbb{Z}_{\ge0}^n$ be a ...
- 16.9k
1
vote
2
answers
410
views
Has this kind of question in topology a special name?
Consider a space $X$ and the group $Homeo(X)/\sim$ of homeomorphisms on $X$ modulo homotopies which are homeomorphisms in each step. One could also consider diffeomorphisms on $X$ or whatsoever.
Have ...
- 11
26
votes
3
answers
1k
views
Proving that a function's image contains (1/n,...,1/n)
This question is a follow-up to a previous question answered by Neil Strickland:
Map from simplex to itself that preserves sub-simplices
Let $B$ denote the closed unit ball in $\mathbb{R}^2$ and let ...
- 645
2
votes
1
answer
4k
views
Sigma Algebra that is not a topology [closed]
Is there an example of a sigma algebra that is not a topology? If this is not the case, is it possible to prove that all sigma algebras are topologies?
23
votes
3
answers
2k
views
An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request
There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter.
But there's ...
- 27.7k
1
vote
1
answer
333
views
Do outer regular outer measures always measure open sets?
Let $ \; \langle X,\mathcal{T} \hspace{.06 in} \rangle \; $ be a second-countable Hausdorff space.
Let $ \; \phi : 2^X \to [0,+\infty] \; $ be an outer regular outer measure.
Does it follow ...
user5810
6
votes
1
answer
678
views
Is it possible to define a closure operator in terms of partial ordering?
For boolean algebra, let's take Roman Sikorski's Boolean Algebras as our reference. After giving a set of axioms, he proves (p.9) that the join of A and B is the least element of the algebra such that ...
- 327
2
votes
1
answer
492
views
Scott topology, but for graphs
Would it be possible to define an order theoretic topology on graphs? I am thinking about the Scott topology. There would be an associated continuous map on graphs.
- 1,313
11
votes
1
answer
949
views
Magma "actions" (or alternatively, "What is the Yoneda lemma for magmas?")
Arguably the most import thing about groups, semigroups and more generally categories, is that they can act on sets (or even collections of sets in the case of a category). This is the basis for all ...
- 2,392
3
votes
0
answers
277
views
For METRIZABLE spaces, do the Banach classes and Baire classes coincide?
In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...
- 456
5
votes
2
answers
739
views
A subcategory of top where subspaces and subobjects coincide?
I think that my question is easily answerable. The question is: What is a nice subcategory of topological spaces where the subobjects are subspaces. I would like the category of compactly generated ...
- 1,325
6
votes
1
answer
517
views
Growth zeta-functions of regular languages
Dear All,
my following question may be known and ought to be known, so in case it is folklore please could you give me the references.
To start, it is obvious that growth of rational languages are ...
- 1,437
3
votes
1
answer
477
views
Equilibria Exist in Compact Convex Forward-Invariant Sets
Theorem. Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and suppose that the autonomous dynamical system $\dot{x} = f(x)$ has a semiflow $\varphi : {\mathbb{R}}_{\geq{0}}...
- 105
1
vote
0
answers
150
views
Follow up question on the measure of the difference between a partial selector and a selector...
This is a different question from my previous question Difference between a partial selector and a selector, however I am going to repeat the preamble...
In Kharazishvili's "Nonmeasurable Sets and ...
- 463
2
votes
1
answer
387
views
Embeddings of vector spaces
Let $V$ be an $n$-dimensional vector space. Is the space of embeddings
$$
\coprod_1^{k} V \to V
$$
path connected for large enough $n$? Clearly $n=1$ is not enough, but I feel like $n=2$ is enough ...
- 569
1
vote
1
answer
636
views
Does anyone know an example of non-separable $L^1$ of a probability space?
It is easy to give examples of non-separable $L^p$ spaces by considering a measure on a big space. If one adds the condition that the space has to have total measure 1, the problem is not as easy.
...
- 96
4
votes
1
answer
860
views
Does pushforward preserve outer regularity?
(ZF + Countable Choice)
Let $\langle A,\mathcal{S} \hspace{.02 in} \rangle$ and $\langle B,\mathcal{T} \hspace{.06 in} \rangle$ be second-countable Hausdorff spaces.
Let $\Sigma$ be a sigma-algebra ...
user5810
0
votes
1
answer
194
views
Difference between a partial selector and a selector...
In Kharazishvili's "Nonmeasurable Sets and Functions" there is the following theorem:
There exists a subset $X$ of $\mathbb{R}$ which is a Vitali set and a Bernstein set.
The proof is as follows:
...
- 463
1
vote
2
answers
405
views
Cardinality of the set of countable dense subgroups of the reals up to isomorphism.
Joel David Hamkins in an answer to my question Countable Dense Sub-Groups of the Reals points out that "one can find an uncountable chain of countable dense additive subgroups of $\mathbb{R}$ whose ...
- 463
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
4
votes
1
answer
240
views
Transversals to singular subvarieties
Say $\mathbb{C}^d \subset Y^{N-k} \subset \mathbb{C}^N$ are closed imbeddings of complex analytic subvarieties of the indicated dimension, $Y$ is not smooth. At a point $y \in Y$, a generic, ...
- 8,723
1
vote
1
answer
374
views
Weak convergence of measures on non-metrizable spaces
(ZF + Countable Choice)
Let $\langle X,\mathcal{T} \hspace{.06 in} \rangle$ be a second-countable Hausdorff space. Let $\mu$ be a Borel measure on $X$.
Let $\langle I,\leq_I \rangle$ be a directed ...
user5810
4
votes
1
answer
436
views
Weak homotopy equivalence of $H$-spaces
Suppose I have an $H$-space $H$ and a topological group $G$, such that for compact spaces $X$ there is a natural equivalence of group valued functors
$[X, H] \to [X, G]$
Now $H$ is a (non-finite) CW-...
- 7,637
5
votes
0
answers
204
views
Shrinking Group Actions
This is a repost from stackexchange. I didn't get an answer, so I figured I'd ask it here.
Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a ...
- 2,751
13
votes
3
answers
1k
views
Map from simplex to itself that preserves sub-simplices
I believe this may be a standard algebraic topology problem, so I apologize in advance if this belongs in stackexchange (it's not a homework problem, however, and came about in a research context). I'...
- 645
6
votes
1
answer
968
views
Lifting local compactness to a covering space
(I decided to repost this from MathSE, since the question seems to not be as easy as I had thought)
NB: In this question, local compactness is used in its weak form, i.e. in a locally compact space, ...
- 2,389
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
9
votes
0
answers
760
views
Characterization of Unusual Topologies of $\mathbb R$
Following some argument over a question on math.SE, I began to wonder:
We all know that in the standard topology there are no continuous bijections from $[0,1]$ to $(0,1)$ (for example by arguments ...
- 39.8k
11
votes
3
answers
832
views
Connectifications?
Like many of my questions, this question is actually aimed at $p$-adic analysis.
One of the main obstacles in doing analysis $p$-adically ist that the $\mathbb{Q}_p$ is totally disconnected.
From ...
- 2,810
5
votes
2
answers
754
views
Do all finitely generated nilpotent semigroups have polynomial growth?
The notion of nilpotency passes nicely from groups to semigroups. Define $q_1(x,y)=xy$ and $$q_{i+1}(x,y,z_1,\cdots,z_i)=q_i(x,y,z_1,\cdots,z_{i-1})z_iq_i(y,x,z_1,\cdots,z_{i-1})$$ inductively for all ...
- 1,437
7
votes
1
answer
433
views
Powers of maps on finite sets
Let $X_n$ be a set with $n$ elements. Write $F(X_n,X_n)$ for the set of maps from $X_n$ to itself. It is a monoid under the operation of composition. Let $m$ be a positive integer. How many maps in $...
- 385
13
votes
2
answers
1k
views
Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
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
4
answers
2k
views
Topological spaces, uncountable subsets and separability
Hi, the following is a well known theorem
Let $M$ be a metric space. If every uncountable subset of $M$ has a limit point, then $M$ is separable.
Question: Is there a similar result for topological ...
- 231
4
votes
3
answers
1k
views
cayley transform for non-square matrices
Hi,
I am optimizing a function over a matrix $U$, where $U \in \mathbb{R}^{m \times n}$ and $U^TU = I$. I do not want to run a constrained maximization program, since employing the constraint $U^TU = ...
- 263
2
votes
0
answers
199
views
Finite topological dimension x local compactness
Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional ...
- 4,729
36
votes
2
answers
2k
views
Can non-homeomorphic spaces have homeomorphic squares?
I an wondering if there are non-homeomorphic spaces $X$ and $Y$ such that $X^2$ is homeomorphic to $Y^2$.
- 531
4
votes
1
answer
240
views
Characterisation of paracompact spaces by some sort of embeddability?
This question was inspired by this question.
Before I start, I don't really mean embedding in what follows. I'm tempted to use plongement, for an exotic touch, but well, that's just a rose by another ...
- 35.5k
28
votes
6
answers
9k
views
Why the triangle inequality?
[Maybe this is asking to be closed; but I thought I'd risk it.]
A metric satisfies the axioms:
$d(x,y)=0$ if and only if $x=y$.
$d(x,y) = d(y,x)$.
$d(x,y) \leq d(x,z) + d(z,y)$.
Similarly (and ...
2
votes
1
answer
243
views
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
13
votes
0
answers
1k
views
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
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