All Questions
5,184 questions
7
votes
2
answers
2k
views
Intersection form in twisted homology (homology with local coefficients)
The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...
- 1,129
17
votes
1
answer
2k
views
Which Fréchet manifolds have a smooth partition of unity?
A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is:
Which Fréchet manifolds have a smooth partition of unity?
How is the ...
- 4,519
82
votes
5
answers
6k
views
How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?
This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
...
- 236k
11
votes
5
answers
1k
views
Confusion over a point in basic category theory
"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological ...
- 1,207
30
votes
8
answers
3k
views
Cryptomorphisms
I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are
Topological Spaces. These can be defined in terms of open sets, ...
4
votes
1
answer
417
views
"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
2
votes
2
answers
1k
views
Simple question of topological cofibration
I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the ...
- 367
32
votes
3
answers
4k
views
Which spaces are inverse limits of discrete spaces ?
There is the following theorem:
"A space $X$ is the inverse limit of a system of discrete finite spaces, if and only if $X$ is totally disconnected, compact and Hausdorff."
A finite discrete space ...
- 11.1k
13
votes
2
answers
713
views
How do you compute the space of lifts of an E-infinity map?
Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g:...
- 3,103
2
votes
1
answer
336
views
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 $\...
6
votes
3
answers
372
views
Notion of finite dimensional simplicial space
I was wondering, what a $N$-dimensional simplicial space $X$ should be. Of course the degeneracy maps force the spaces to be nonempty in high dimensions. Currently I have two different versions and i ...
- 11.1k
1
vote
1
answer
190
views
Is the semigroup M(n, Z) finitely presented? If so, where can I find a presentation of it?
I am new to semigroup research, so I apologize if this is an easy question.
- 87
13
votes
4
answers
1k
views
Why is a monoid with closed symmetric monoidal module category commutative?
Given a symmetric monoidal category and a monoid object A in it, one can form the category of modules over this monoid object, i.e. objects are $A \otimes M \rightarrow M$ satisfying the natural ...
- 12.3k
38
votes
7
answers
5k
views
Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)
In this question, Harry Gindi states:
The fact that a commutative ring has a natural topological space associated with it is a really interesting coincidence.
Moreover, in the answers, Pete L. ...
- 21k
7
votes
3
answers
1k
views
The continuous as the limit of the discrete
Reading this documment: www.math.ucla.edu/~tao/preprints/compactness.pdf, I got interested in the following thing: "One can also use compactifications to view the continuous as the limit of the ...
- 153
52
votes
7
answers
8k
views
"Algebraic" topologies like the Zariski topology?
The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...
2
votes
1
answer
1k
views
monoid ring and some structure within it - how is it called?
I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure ...
- 1,626
-3
votes
2
answers
1k
views
Finite versus infinite on non-Hausdorff topologies [closed]
Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a ...
- 283
1
vote
2
answers
394
views
Relations in matrix semigroups
Suppose that $A_1, \dots, A_k \in M_n(\mathbb{Q})$ and $S$ is the semigroup generated by them. Two questions: are there always a finite set of relations $\{R_i\}$ among the $A_j$ such that $S$ is ...
- 4,646
11
votes
6
answers
1k
views
Computing the structure of the group completion of an abelian monoid, how hard can it be?
Cherry Kearton, Bayer-Fluckiger and others have results that say the monoid of isotopy classes of smooth oriented embeddings of $S^n$ in $S^{n+2}$ is not a free commutative monoid provided $n \geq 3$. ...
- 44.4k
7
votes
2
answers
370
views
Wants: Polynomial Time Algorithm for Decomposing a Multiset of Rationals into Two Additive Subsets.
First, allow me to say that this problem was posed to me by a professor in the department. It is related to his research in a way that I do not know. However, since I couldn't come up with anything ...
- 4,842
13
votes
2
answers
646
views
Functions separting points in Hausdorff spaces
A colleague in algebra asked me this, and I couldn't answer it. On the Wikipedia page for "epimorphism" it is claimed that in the category of Hausdorff spaces and continuous maps, a function is epi ...
- 18.7k
5
votes
1
answer
723
views
Sheaf condition and representability in the category Top
This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category $Funct(\mathcal{C}^{...
- 7,442
4
votes
2
answers
551
views
Normality of an affine semigroup
An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is ...
- 111
3
votes
6
answers
3k
views
Cone in a metric space
We know the definition of a cone in a Real Banach Space.
I want to know if there is any definition for a cone in an abstract metric space.
Have you ever seen such definition anywhere?
- 520
7
votes
1
answer
3k
views
definition of the end of a manifold?
I was hoping if somebody could help me out with the terminology. I've found that the "end of a manifold" is a function assigning to each compact set K a conected component e(K) of the ...
- 71
2
votes
3
answers
1k
views
Baire category theorem
Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = \cup_{n=1}^\infty A_n$. Let $\bar{A}$ denote the closure of $A$.
Let's call the following statement (2): ...
- 498
26
votes
4
answers
4k
views
What is the "right" universal property of the completion of a metric space?
I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes:
Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...
- 65.4k
6
votes
2
answers
657
views
Properties of the class of topological spaces possessing a CW-structure
Let ${\mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure).
Is it true that for a covering map $E\stackrel{f}{\to} ...
- 2,756
3
votes
3
answers
728
views
What do you call the product of a circle and an annulus?
What would you call the product of an annulus and $S^1$ (a 'thickened' torus like 3-manifold)?
More generally, is there an archive or list online of names assigned to various (non-standard) manifolds ...
- 3,165
13
votes
4
answers
1k
views
nonhausdorff dimension
if $X$ is a topological space, a first step in making $X$ hausdorff is taking the quotient $H(X)=X/\sim$, where $\sim$ is the equivalence relation generated by: if $x,y$ cannot be seperated by ...
- 63.1k
9
votes
2
answers
934
views
Space whose product with paracompact space is paracompact
Is there a nice characterization of topological spaces with the property that the product with any paracompact space is paracompact?
All compact spaces have this property (this can be shown from the ...
- 7,320
13
votes
3
answers
978
views
Model Structure/Homotopy Pushouts in topological monoids?
Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$.
Can the model category structure on $\textsf{Top}$ (Serre fibrations, ...
- 1,033
47
votes
4
answers
4k
views
Which topological spaces admit a nonstandard metric?
My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric.
That is, let us define that a topological ...
- 236k
31
votes
4
answers
5k
views
Are all Hawaiian Earrings homeomorphic?
The Hawaiian Earring is usually constructed as the union of circles of radius 1/n centered at (0,1/n): $\bigcup_1^\infty \left[ (0, \frac{1}{n}) + \frac{1}{n}S^1 \right]$. However, nothing stops us ...
- 22.8k
4
votes
3
answers
1k
views
Morse theory and Euler characteristics
Suppose we have a space M with a real-valued, differentiable function F on M. Under what conditions on F will the Euler characteristic of M be expressed as a (signed) sum of Euler characteristics of ...
- 1,129
5
votes
2
answers
1k
views
Relating Euler characteristic, intersection product, Morse theory (plus SU(2) and 3-manifolds)
Suppose we have a (closed, oriented) 3-manifold M with a Heegard surface F of genus g. Let F* denote F with a puncture. Then the space H of representations of pi_1(F*) on SU(2) is just SU(2)^2g, and ...
- 1,129
6
votes
1
answer
254
views
p-adic noninvariance of dimension
Let $p$ be a prime number. Let $n,m \geq 1$ be such that the topological spaces $\mathbb{Q}_p^n$ and $\mathbb{Q}_p^m$ are homeomorphic. Can we conclude $n=m$?
For $\mathbb{Z}_p$ it's false: In fact, ...
- 63.1k
14
votes
1
answer
2k
views
When are epimorphisms of algebraic objects surjective?
Let $C$ be the category of $\tau$-algebras for some type $\tau$. Consider the statements:
Every monomorphism is regular.
Every epimorphism in $C$ is surjective.
It is easy to see that 1. implies 2. ...
- 63.1k
13
votes
6
answers
3k
views
When does local invertibility imply invertibility?
Generally, local invertibility does not imply invertibility. However, for differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ then surjectivity and local invertibility do imply invertibility.
...
- 26.8k
37
votes
14
answers
5k
views
What are interesting families of subsets of a given set?
Motivation
The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$.
Indeed, one defines a topology on $S$ to be a family of subsets ...
- 33.3k
-4
votes
1
answer
8k
views
How to transform a plane into a sphere? [SOLVED] [closed]
Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into ...
- 555
24
votes
2
answers
1k
views
Which are the rigid suborders of the real line?
Which are the rigid suborders of the real line?
If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure ...
- 236k
5
votes
1
answer
968
views
Can topologies induce a metric? (revised)
This is a revised version of a question I already posted, but which patently was ill posed. Please give me another try.
For comparison's sake, the axioms of a metric:
Axiom A1: $(\forall x)\ d(x,x) =...
- 9,720
9
votes
9
answers
4k
views
Help me with this proof: Drop a printed map of the land on the land and there must be some common point.
Hi, I have a minor in math and this is not a homework problem - my prof mentioned it 5 years ago and I could not even begin to tackle it until I took a good intro to linear algebra (after work). ...
- 171
36
votes
4
answers
4k
views
How far is Lindelöf from compactness?
A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the ...
- 2,396
13
votes
1
answer
766
views
Is Top_4 (normal spaces) a reflective subcategory of Top_3 (regular spaces)?
I’m studying some category theory by reading Mac Lane linearly and solving exercises.
In question 5.9.4 of the second edition, the reader is asked to construct left adjoints for each of the inclusion ...
- 1,411
39
votes
3
answers
6k
views
Why do finite homotopy groups imply finite homology groups?
Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\...
- 2,782
1
vote
1
answer
164
views
The proper name for a kind of ordered space [closed]
I'm trying to find the correct term for a specific kind of totally ordered space:
Let $S$ be a totally ordered space with strict total order $<$.
Property: For any two $s_{1}$ and $s_{2}$ in $S$ ...
- 121
-2
votes
2
answers
931
views
Can topologies induce a metric?
Let {X,T} be a topology, T the set of open subsets of X.
Definition: Three points x, y, z of X are in relation N (Nxyz, read "x is nearer to y than to z") iff
there is a basis B of T and b in B ...
- 9,720