All Questions
5,183 questions
38
votes
5
answers
4k
views
When factors may be cancelled in homeomorphic products?
It is easy to see that if $A\times B$ is homeomorphic to $A\times C$ for topological spaces $A$, $B$, $C$, then one may not conclude that $B$ and $C$ are homeomorphic (for example, take $C=B^2$, $A=B^{...
- 109k
22
votes
8
answers
3k
views
Connections between ultrafilters in topology and logic
I have a some-what vague question. It seems to me that there are two main ways in which ultrafilters (on a set) can be used. One is in topology. The notion of an ultrafilter converging to a point is ...
- 15.5k
1
vote
1
answer
716
views
An example of a space which is locally relatively contractible but not contractible?
A space $X$ is called locally contractible it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the ...
- 35.5k
18
votes
4
answers
4k
views
Why are topological ideas so important in arithmetic?
For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in ...
- 4,351
54
votes
4
answers
6k
views
Are the rationals homeomorphic to any power of the rationals?
I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ (...
- 11.1k
23
votes
3
answers
9k
views
Sets with positive Lebesgue measure boundary
Consider a compact subset $K$ of $R^n$ which is the closure of its interior. Does its boundary $\partial K$ have zero Lebesgue measure ?
I guess it's wrong, because the topological assumption is ...
- 18.7k
17
votes
6
answers
24k
views
How to understand the concept of compact space [closed]
the definition of compact space is: A subset K of a metric space X is said to be compact if every open cover of K contains finite subcovers. What is the meaning of defining a space is "compact". I ...
- 181
7
votes
4
answers
1k
views
Torsors for monoids
Torsors are defined as a special kind of group action. I am wondering whether the analogous notion exists for monoid actions. Some references would be helpful.
In general I'm interesting in the ...
- 1,882
2
votes
2
answers
2k
views
Maximum number of shortest-paths
I would like to know if there is a equation for the maximum number of shortest paths that pass through r where r is a node contained in any path from node s (a fixed node, i mean, s is the only source ...
- 23
5
votes
1
answer
1k
views
Do continuous maps give continuity in the 'topology' of Hausdorff distance?
I was reading this question:
limiting behaviour of converging loops on a torus
And I wanted to be able to give an argument along the lines of: "If your loops are converging in your torus, their ...
- 3,230
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
20
votes
3
answers
1k
views
How thinly connected can a closed subset of Hilbert space be?
Let H be a separable (and infinite-dimensional) Hilbert space. Is it known whether there exists an infinite
subset C of H with the following properties.? (1) C is connected and closed in H. (2) No ...
- 6,215
7
votes
1
answer
789
views
Counting submanifolds of the plane
After thinking about this question and reading this one I am led to ask for an uncountable collection of homeomorphism types of boundaryless connected path-connected submanifolds of the plane.
My ...
- 28.2k
5
votes
1
answer
2k
views
Algorithms for the Lakes of Wada
The Lakes of Wada partitions the unit square in to three regions, all of whom share a common boundary. The Wikipedia entry (http://en.wikipedia.org/wiki/Lakes_of_Wada) gives a construction approach, ...
- 2,537
21
votes
2
answers
3k
views
Integer matrices with no integer eigenvalues
Let $$A = \begin{pmatrix} 3&1 \\ 0&1 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1&0\\ 1&2 \end{pmatrix}$$ I want to show that the only elements of the semigroup generated by $A$ and $B$...
- 1,045
5
votes
0
answers
350
views
Chain/Hierarchy of Monoids
Let's assume that we have the following collection of structures:
Some space $P$.
Monoids $(M_{i+1},\circ_{i+1})$, and
Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
And $\bullet_{0}:...
- 1,882
32
votes
3
answers
6k
views
Is "compact implies sequentially compact" consistent with ZF?
Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...
- 26.8k
14
votes
4
answers
2k
views
Products of Baire spaces
I could not find any references about this fact. I apologize if this is completely trivial, but is the product of two Baire spaces, or for that matter of finitely many of them a Baire space? Now is a ...
- 3,804
6
votes
2
answers
1k
views
Quantitative questions about the size of a finite epsilon net
Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...
- 943
9
votes
2
answers
1k
views
A question about the Osgood curve
Does every sub-arc of the Osgood curve (with distinct end-points) have positive two-dimensional
Lebesgue measure? If not, do there exist Jordan curves whch have this property?
- 6,215
3
votes
0
answers
267
views
Maps of loop spaces with infinity-bounded differential.
I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally:
In the following piece-wise smooth means smooth on ...
- 2,590
11
votes
2
answers
2k
views
Is there an uncountable, non-discrete, Hausdorff Toronto space?
We call a topological space $X$ a Toronto space if for any subspace $Y \subseteq X$ such that $Y$ and $X$ have the same cardinality it follows that $Y$ is homeomorphic to $X$.
Does anybody know what ...
- 2,035
6
votes
0
answers
2k
views
Weak lower semi-continuity
Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type
$F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will even ...
11
votes
2
answers
1k
views
Why free topological groups on Tychonoff spaces?
This is a question of the motivation for a common assumption found in the literature.
The free topological group $F(X)$ on a space $X$ exists for all spaces $X$ (It seems this was first shown by ...
- 7,193
17
votes
5
answers
830
views
How can one characterise compactness-by-experiment?
There are a myriad different variations on the theme of "compactness", and some of them have even made it on to Wikipedia. I'm interested in finding out more about types of compactness that ...
- 26.8k
4
votes
1
answer
671
views
Sections of an etale space
In R.O.Wells book "Differential Analysis on Complex Manifolds" p. 44 proof of Theorem 2.2 part b) the author claims that any two sections of an etale space which agree at a point agree in some ...
- 43
5
votes
2
answers
878
views
What is an example of a non-regular, totally path-disconnected Hausdorff space?
I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for ...
- 35.5k
37
votes
5
answers
4k
views
Reference for the Gelfand duality theorem for commutative von Neumann algebras
The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent:
(1) The opposite category of the category of commutative von Neumann ...
- 37.8k
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
13
votes
2
answers
659
views
Noncontractible connected topological rings ?
Are there any non-contractible connected topological rings?
Of course, such a thing cannot be a (topological) algebra over the reals.
(I have a vague memory of having a glance at an erticle by Lurie ...
- 23.3k
5
votes
2
answers
1k
views
Improvements of the Baire Category Theorem under (not CH)?
The Baire category theorem implies that a nonempty complete metric space without isolated points must be uncountable. In many situations I have encountered, the "natural examples" of ...
- 65.4k
17
votes
5
answers
5k
views
Why are inverse images more important than images in mathematics?
Why are inverse images of functions more central to mathematics than the image?
I have a sequence of related questions:
Why the fixation on continuous maps as opposed to open maps? (Is there an ...
11
votes
3
answers
1k
views
Which properties of finite simplicial sets can be computed?
A simplicial set $X$ is a a combinatorial model for a topological space $|X|$, its realization, and conversely every topological space is weakly equivalent to such a realization of a simplicial set. I ...
- 727
28
votes
7
answers
13k
views
Regular borel measures on metric spaces
When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
- 18.7k
-5
votes
1
answer
483
views
For every proximity, does there exist a uniformity which generates this proximity?
For every proximity, does there exist a uniformity which generates this proximity?
This question may be generalized for different generalizations of proximities and uniformities. In fact I need it ...
- 765
4
votes
1
answer
675
views
Name for topology making group action continuous
Fix a set $X$ with right $G$-action. Give $X$ a topology $\tau$ and make $G$ a topological group. (These topologies need not make the action continuous).
We can define another topology $\tau'$ on $...
- 2,895
8
votes
3
answers
1k
views
Locally complete space is topologically equivalent to a complete space
Can someone please tell me where I can find a citeable reference for the following result:
Call the metric space $(X, d)$ "locally complete" if for every $x \in X$ there a neighbourhood of $x$ which ...
- 2,895
3
votes
2
answers
699
views
Conditions useful for proving paracompactness
I have a family of properties which I want to show taken together imply paracompactness (I can show that they are all implied by paracompactness). I can prove a whole bunch of things which are ...
- 1,331
5
votes
1
answer
438
views
Fixed points sets of pushouts
Let $G$ be a group and $X \to Y, X \to Z$ morphisms of $G$-sets with pushout $P=Y \cup_X Z$. Is then $P^G$ the pushout of $X^G \to Y^G, X^G \to Z^G$? This is not clear from general category theory, ...
- 63.1k
1
vote
3
answers
995
views
SO(3) knot polynomials
Can one use the real lie algebra so(3) to get knot polynomials? If so, do they have a skein relation (I presume they would, if they come from R-matrices in some standard way. If so, is the R-matrix ...
- 1,129
0
votes
2
answers
172
views
small extensions of the free semigroup of rank 1
Let N denote the free semigroup of rank 1. Say that a semigroup T is a small extension
of N if N embeds in T and |T - N| is finite. Is there some kind of classification
of small extensions of N? ...
- 91
34
votes
2
answers
3k
views
"Transitivity" of the Stone-Cech compactification
Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \setminus \mathbb{N}$ be two non-principal elements of this ...
- 114k
3
votes
0
answers
721
views
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
2
votes
5
answers
1k
views
Is it true that the only interesting topologies are metric topologies and weak topologies?
In "Infinite dimensional analysis, A hitchhikers guide" by Aliprantis and Border, they write that these 2 classes of topologies "by and large include everything of interest".
@Pete Clarke: I was ...
- 4,351
4
votes
2
answers
1k
views
Is it still impossible to partition the plane into Jordan curves without choice?
It is an easy exercise to show that the Euclidean plane cannot be partitioned into round circles (note however that it is possible to do so for $\mathbb{R}^3$). It seems almost obvious that it is not ...
- 14.4k
48
votes
3
answers
13k
views
When is a Homology Class Represented by a Submanifold? [duplicate]
Possible Duplicate:
Cohomology and fundamental classes
Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class $\phi_*[N]\in H_*(M)$ ...
- 2,283
15
votes
5
answers
3k
views
Is the pure braid group on three strands generated as a normal subgroup of the braid group by the six-crossing braid?
Artin's presentation of braid group on three strands is:
$$ B_3 = \langle l,r : lrl = rlr \rangle $$
where you should think of "$l$" as the positive crossing between the left and middle strands and "$...
- 54.6k
3
votes
3
answers
444
views
Shape of long sequences in C(ω_1)
Apologies for the vague title - I couldn't come up with a single sentence that summarised this problem well. If you can, please edit or suggest a better one!
This question is also rather specific and ...
- 1,331
8
votes
4
answers
1k
views
Does the set of open sets in a topological space have a topology itself?
If X is a topological space, and A consists of all of X's open sets, can we define a natural topology on A (using the topology of X)?
- 165
6
votes
1
answer
297
views
Is there a "natural" characterization of when X × βN is normal?
As per a recent question of mine, $\omega_1 \times \beta \mathbb{N}$ is not normal. I'm wondering whether there's some sort of "natural" condition that describes when a space has a normal product with ...
- 1,331