All Questions
5,184 questions
2
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Spectral sequences in Hypercohomology of sheaves
Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this ...
- 360
2
votes
1
answer
689
views
Partitions of an interval
This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there.
Specifically, consider "partitions" ...
- 2,358
16
votes
3
answers
991
views
Where else do the (topology) separation axioms turn up?
As an undergraduate I learned point-set topology from Munkres's book, as did many others.
One topic that gets a lot of attention is the separation axioms. For example, a space $X$ is normal if any ...
- 7,347
30
votes
4
answers
2k
views
is f a polynomial provided that it is "partially" smooth?
Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in ...
- 407
2
votes
1
answer
296
views
Does every ultrafilter has single limit imply Hausdorff separation
If a topological space $X$ enjoys the property that every ultrafilter $U$ on $X$ has a single limit, must $X$ be a Hausdorff space?
(Ultrafilters here consist of arbitrary subsets (so not necessarily,...
- 427
4
votes
3
answers
480
views
closed meagre sets
A closed meagre subset of $[0,1]$ is either countable or homeomorphic to the Cantor set: either way it is $0$-dimensional.
Q.1. Is every closed meagre subset of an $n$-dimensional locally compact ...
- 607
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
16
votes
2
answers
2k
views
Is Stone-Čech compactification of 0-dimensional space also 0-dimensional?
What is an example of a 0-dimensional locally compact Hausdorff space $X$ for which the Stone-Čech compactification $\beta(X)$ is not 0-dimensional?
It is known that if $X$ is a 0-dimensional locally ...
- 1,704
7
votes
4
answers
3k
views
What is a good application of Urysohn's Theorem?
Urysohn's Metrization Theorem states that every Hausdorff second-countable regular space is metrizable.
What is an example of a Hausdorff second-countable regular space where it is difficult to prove ...
- 3,284
6
votes
2
answers
2k
views
Condition to ensure that the product of closed maps be closed
If $f_i : X_i \to Y_i$ with $i=1,2,\ldots,n$ are closed maps between topological space it is known that their product map
$$f : X_1 \times \cdots \times X_n \to Y_1 \times \cdots \times Y_n : (x_1, \...
- 463
11
votes
2
answers
2k
views
When is the connected sum of manifolds orientation-independent?
Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed?
If $N$ ...
- 931
3
votes
2
answers
447
views
Number of non-intersecting non-homotopic simple closed curve
How many simple closed curves can be put on a orientable genus $g$ surface $\Sigma_g$ such that the following are true:
The curves are pairwise non-homotopic
The curves are pairwise set-theoretically ...
user13946
8
votes
2
answers
427
views
Is there a general result that theorems about finite structures proved in ZFC can be proved in ZF?
The title question is too vague so let me be specific.
Much of modern finite semigroup theory uses profinite semigroups and properties of profinite semigroups that depend on the existence of prime ...
- 38.6k
0
votes
0
answers
635
views
Do homotopic non-intersecting simple closed curves separate the surface?
Let $C_1$ and $C_2$ be two simple closed curves on an orientable compact surface $S$, such that:
They are homotopic to each other.
They are set-theoretically disjoint.
Is $S\setminus(C_1 \cup C_2)$ ...
user13946
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
1
vote
2
answers
1k
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Hypercohomology of a complex of sheaves that might be acyclic (or might not)
Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves
...
- 360
2
votes
2
answers
439
views
countably complete filters
Is there any description of the set of countably complete filters on the lattice of dense $G_{\delta}$ subsets of a compact, second countable metric space? [I haven't just dreamt this up: it describes ...
- 607
5
votes
3
answers
718
views
Subsets of $\mathbb{R}^+$ closed under addition
No one's answered the question cumulant problem so here's a simpler question: Has anyone described or catalogued all sets of non-negative real numbers that are closed under addition? In particular, ...
8
votes
1
answer
938
views
Filling $\mathbb{R}^3$ with skew lines
I would like to know if it is possible to fill $\mathbb{R}^3$ with lines with the
following two properties:
(1) Every point $x \in \mathbb{R}^3$ is contained in precisely one line.
(2) Every ...
- 151k
7
votes
0
answers
517
views
Is there a natural topology on the set of open sets ?
Given a topological space $(X,\mathcal{O})$ can one assign a natural topology to $\mathcal{O}$ such that
1) The intersection of a compact set of open sets is again open,
2) The maps $\cap,\cup:\...
- 11.1k
8
votes
2
answers
826
views
Which semigroups can be linearly ordered?
As usual I consider a semigroup to be a structure $(A, +)$ such that $+$ is an associative binary function over the set $A$. The notion of linearly-ordered semigroup corresponds to structures of the ...
- 768
6
votes
2
answers
257
views
Borel functions on $\omega_1$
Endow $\omega_1$ with order topology. It is easy to show that each continuous function $f\colon \omega_1\to \mathbb{R}$ is eventually constant. Is the same true for Borel functions?
- 61
50
votes
7
answers
5k
views
Is there an algebraic approach to metric spaces?
It is well known that most topological spaces can be studied via their algebra of continuous real-valued (or complex-valued) functions. For instance, in the setting of compact Hausdorff spaces, there ...
- 4,874
2
votes
0
answers
121
views
Graphs, non-Hausdorfness and Wallman compactifications of non-regular spaces
Given a non-Hausdorff space $X$, one can form a graph $G_X$: vertices the points of $X$, edges indicating point pairs not separated by open sets. Up to graph-theoretically (but not topologically) ...
- 17.6k
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
2
votes
1
answer
214
views
union of Stone-Cech remainders
Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [...
- 607
1
vote
2
answers
443
views
submonoids of Z_n
Anyone knows how to describe explicitly the submonoids of Z_n, regarded as a multiplicative
monoid?
13
votes
1
answer
545
views
Square of a continuous map
Recently a student asked me the following (elementary looking) question :
If $T$ is an invertible linear transformation of some finite-dimensional space $E$ into itself which factorizes as $T = f \...
- 7,249
11
votes
3
answers
942
views
What is the smallest variety of algebras containing all fields?
A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
- 2,464
20
votes
1
answer
994
views
Which spaces are characterized by functions with compact support ?
It's well known that two locally compact Hausdorff spaces $X, Y$ are homeomorphic iff the rings $C_0(X), C_0(Y)$ (continuous functions vanishing at infinity) are isomorphic.
Is there a class $\...
- 16.2k
4
votes
1
answer
216
views
closed set and z-ultrafilter on normal space
Let $X$ be a completely regular, Hausdorff topological space and let $\cal F$ be a $z$-ultrafilter on $X$. Then for each zero set $W$ in $X$, either $W\in \cal F$ or there exists $Z\in \cal F$ such ...
- 607
10
votes
3
answers
2k
views
Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 360
23
votes
4
answers
1k
views
Spaces with no topological monoid structure which are homotopy equivalent to topological monoids
In motivating $A_\infty$-spaces to my students I'm going to insist on the homotopy invariance of the notion, saying that "being $A_\infty$ is the homotopy invariant version of being a topological ...
- 6,777
36
votes
2
answers
3k
views
Computing self-intersections with complex analysis
It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...
- 22.8k
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
5
votes
1
answer
452
views
Least cardinality of a set of points in the plane
What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all ...
- 98
6
votes
1
answer
405
views
Infinite closed partition of the real line with no closed infinite unions
Is there a partition of the real line into infinitely many closed subsets so that no infinite union of these subsets (except the whole space) is closed?
This question was asked also at math....
- 367
9
votes
2
answers
2k
views
Stone-Čech compactification of $\mathbb R$
Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it ...
- 385
22
votes
2
answers
977
views
Mapping from a finite index subgroup onto the whole group
Dear All,
here is the question:
Does there exist a finitely generated group $G$ with a proper subgroup $H$ of finite index, and an (onto) homomorphism $\phi:G\to G$ such that $\phi(H)=G$?
My guess ...
- 1,437
11
votes
1
answer
1k
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Reference request for TQFT, functoriality
I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.
It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;
The function $(M, \partial_{-}M, \...
- 111
3
votes
1
answer
524
views
Metrizable implies hemicompact
In the paper
R. Arens: A Topology for Spaces of Transformations, Ann. of Math. 47(1946), 480-495
the author states in the introduction that if $B$ is a metric space and the space of continuous ...
- 16.2k
4
votes
2
answers
783
views
Moore path space.
Let $X$ a topological space and $MX$ the Moore path space of $X$
there is two maps from $\alpha,\omega: MX\rightarrow X$ (evaluation in 0 and evaluation at the total length).
The classical path ...
- 1,974
6
votes
0
answers
715
views
What is the structure of a space of $\sigma$-algebras?
Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
- 8,512
2
votes
2
answers
704
views
Topology on Set of Prime Filters of a Distributive Lattice
Given a distributive lattice $A$ we can look at $Spec(A)$, whose points are prime ideals and its open sets are given similarly to the Zariski topology on Spec of a ring. That is, the basis of open ...
- 10.4k
5
votes
1
answer
2k
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Proof that the Pontryagin dual of a topological group is a topological group
I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* \...
- 285
5
votes
0
answers
135
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Possible homogeneity of infinite dimensional Sierpinski carpet analogues?
Start with the Hilbert cube $H=I^\omega$, thinking of its coordinates as written in ternary expansion.
Construct subsets $S_n$ by removing points from $H$ if for any $m$,
at least $n$ of the ...
- 17.6k
6
votes
1
answer
815
views
When is a Topological pushout also a Smooth pushout?
I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:
Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram ...
- 732
4
votes
1
answer
2k
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Connected level sets
This may be an ill-posed question, but suppose I have a collection of continuous, bounded, scalar-valued nonnegative functions $f_1(x,y),\dots,f_n(x.y)$ defined on the closed unit disk. Given a ...
- 645
4
votes
0
answers
223
views
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
5
votes
1
answer
304
views
flat maps of monoids which are not localizations
It is well known that a localization $S^{-1}R$ of a commutative ring $R$ is flat as a $R$-module.
Rather, I am looking for extensions of rings which share certain properties of localizations, like ...
- 6,210