All Questions
5,183 questions
0
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10
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What is an explicit example of a sequence converging to two different points? [closed]
In principle a sequence in a non-Hausdorff space can converge to two points simultaneously.
Can anyone give me an explicit example of the above?
Or tell me any method of generating such kinds of ...
- 3,541
3
votes
3
answers
1k
views
Countable atomless boolean algebra covered by a larger boolean algebra
Suppose $Q$ is an atomless countable boolean algebra, and $B$ is an arbitrary atomless boolean algebra. $Q$ is unique modulo isomorphisms. There is a subalgebra in $B$ that is isomorphic to $Q$. There ...
- 355
9
votes
2
answers
4k
views
Quotient of a Hausdorff topological group by a closed subgroup
Sorry if this question is below the level of this site: I've read that the quotient of a Hausdorff topological group by a closed subgroup is again Hausdorff. I've thought about it but can't seem to ...
- 1,479
2
votes
1
answer
510
views
Are the C(S^n, S^n)'s homeomorphic ?
Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ?
[both endowed with the sup metric (or equivalently the compact-open topology)]
Generally, C(S^n, S^n), with n >= 1, is a ...
- 4,060
26
votes
15
answers
19k
views
Learning Topology
EDIT (Harry): Since this question in its original form was poorly stated (asked about topology rather than graph theory), but we have a list of Topology books in the answers, I guess you should go ...
5
votes
2
answers
521
views
Freeing a sphere from within a sphere
We can embed $S^2\times I$ into $\mathbb{R}^3$ by taking a compact 3-ball and removing an open 3-ball from its interior. Taking the boundary gives an embedding $i: S^2\sqcup S^2\hookrightarrow\mathbb{...
4
votes
3
answers
609
views
When is $A : C(X) \to C(Y)$ a composition operator?
A composition operator $C\_T : C(X) \to C(Y)$ with $T \in C(Y, X)$ is defined by $C\_T f := f \circ T, f \in C(X)$.
I read in the book about Composition Operators by Singh and others that a ...
- 1,338
14
votes
3
answers
1k
views
What is a monoidal metric space?
At time of writing, the highest rated answer to my question What is a metric space? is Tom Leinster's account of Lawvere's description of a metric space as an enriched category. This prompted my ...
- 26.8k
3
votes
3
answers
447
views
Representations of finite commutative band semigroups
I think it's clear that commutative semigroups S that are also bands, i.e. $e^2 = e$ for all e, correspond to finite posets (consider the elements of the semigroups as sets, where the intersection of ...
- 2,216
12
votes
2
answers
2k
views
Topological Rings
Is it true that, if S is a subring of a separable topological Noetherian ring R,
then S is separable, too ?
- 4,060
23
votes
4
answers
2k
views
Which is the correct ring of functions for a topological space?
There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one.
...
- 54.6k
8
votes
1
answer
655
views
Coherent spaces
In Proofs and Types, Girard discusses coherent (or coherence) spaces, which is defined as a set family which is closed downward ($a\in A,b\subseteq a\Rightarrow b\in A$), and binary complete (If $M\...
- 475
11
votes
2
answers
61k
views
Difference between connected vs strongly connected vs complete graphs [closed]
What is the difference between
connected
strongly-connected and
complete?
My understanding is:
connected: you can get to every vertex from every other vertex.
strongly connected: every vertex ...
- 219
3
votes
1
answer
361
views
Is the coproduct of fibrant spectra fibrant again?
Define an $S^{1}$-spectrum $E$ to be a sequence of pointed simplicial sets $E_{n},\\ n=0,1,2...$ with assembly morphisms $\sigma_{n}:S^{1}\wedge E_{n}\rightarrow E_{n+1}$.
An $S^{1}$-spectrum $E$ is ...
- 51
9
votes
5
answers
1k
views
References/literature for pushouts in category of commutative monoids? [ed. - amalgams]
This is more of a request for pointers to relevant literature than a question per se. I am, erm, looking at a paper which uses a kind of iterated pushout construction to obtain a commutative monoid ...
- 25.8k
2
votes
2
answers
1k
views
When is a Hausdorff space metrisable?
This question may be a little too easy for this site, but I'll ask it anyway: when is a Hausdorff topological space metrisable?
- 29
4
votes
1
answer
2k
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Lebesgue measure of boundary of Caccioppoli set
Can anything be said about the measure of the topological boundary of a Cacciopoli set in $R^n$? Of course, the reduced boundary has finite (n-1)-dimensional Hausdorff measure, but this does not say ...
- 320
16
votes
2
answers
1k
views
Which topological spaces have the property that their sheaves of continuous functions are determined by their global sections?
I hope I'm using the terminology correctly. What I mean is this: fix $K = \mathbb{R}$ or $\mathbb{C}$ (I'm interested in both cases). Which topological spaces $X$ have the property that for every ...
- 118k
2
votes
3
answers
369
views
How do we know that a map $f: U \to Y$ extends to $\bar{U}$?
I read the following fact: if $U$ is an open subset of $P_k^1$ and $f: U \to U$ is an automorphism of schemes, then $f$ extends to an automorphism of $P_k^1$. Thus I was curious: is there a general ...
- 25.6k
42
votes
8
answers
5k
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What is a metric space?
According to categorical lore, objects in a category are just a way of separating morphisms. The objects themselves are considered slightly disparagingly. In particular, if I can't distinguish ...
- 26.8k
4
votes
1
answer
448
views
Is there a name for this topology?
Let $X$ be a set and let $f: X\longrightarrow X$ be a function on $X$. Introduce a topology on $X$ by the following basis of open sets: for any subset $S$ of $X$, let $B_S$ be the set of forward ...
- 2,830
5
votes
1
answer
320
views
Ramified covers of S^n
This question has been inspired by covering 3-torus post.
Is it true that any good (smooth, compact, oriented) $n$-manifold can be mapped to $S^n$ in such a way that the map is true covering away ...
- 15.1k
65
votes
14
answers
6k
views
Notions of convergence not corresponding to topologies
This question concerns the ramifications of the following interesting problem that
appeared on Ed Nelson's final exam on Functional Analysis some years ago:
Exam question: Is there a metric on the ...
- 801
3
votes
1
answer
242
views
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
48
votes
8
answers
8k
views
When are there enough projective sheaves on a space X?
This question is being asked on behalf of a colleague of mine.
Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every ...
- 65.4k
6
votes
6
answers
2k
views
Spectra of $C^*$ algebras
Gelfand-Naimark structure theorem for $C^* $ algebras gives a canonical isometric * isomorphism between any commutative unital $C^* $ algebra $A$ and the algebra of continuous complex-valued functions ...
- 2,479
28
votes
8
answers
4k
views
Is there a compact group of countably infinite cardinality?
Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.
Is there a compact (Hausdorff, or even T1) (topological) ...
- 12.6k
4
votes
4
answers
1k
views
Boundary of planar region
Is there a necessary and sufficient condition for the boundary of a planar region to be a finite union of Jordan curves?
- 1,159
8
votes
5
answers
1k
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Examples of left reversible semigroups
I am looking for concrete examples of cancellative, left reversible semigroups. Left reversible semigroups are also called "Ore semigroups". See this wikipedia page for the definition of a left ...
- 550
9
votes
1
answer
625
views
Stable presentable categories as module categories
There is a theorem of Schwede and Shipley which classifies categories of modules over an A∞ ring spectrum as those stable presentable (∞,1)-categories with a compact generator. Suppose I ...
- 25.2k
11
votes
3
answers
892
views
How much "Morse theory" can be accomplished given only a continuous transformation of a space?
If $M$ is a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse-Smale function (which is just a rigorous way to say "generic smooth function"), then Morse theory essentially recovers the manifold ...
- 5,992
2
votes
1
answer
265
views
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
19
votes
4
answers
18k
views
On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...
It is well-known that
A: The series of the reciprocals of the primes diverges
My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers.
Property A ...
- 8,842
60
votes
6
answers
7k
views
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
- 44.4k
5
votes
5
answers
972
views
A walk on a compact 2D surface embedded in 3-space that never returns home
At the risk of asking an uninformed question...
Imagine an ant on a compact two-dimensional surface embedded in 3-space. The ant is placed at a point on the surface with random orientation. Once ...
- 811
2
votes
2
answers
308
views
Convexity Theorem of Hamiltonian actions - the connectedness part
Suppose we have a Hamiltonian action of a torus $T = T^m = R^m/Z^m$ on a compact, connected symplectic manifold $M$. According to the convexity theorem, we know every fiber of the momentum map $\mu: M\...
- 377
2
votes
4
answers
634
views
Can connectedness of fibers of a smooth map be checked on a dense set?
Suppose $f: M\to N$ is a smooth map between two smooth manifolds, with $M$ compact and connected, and suppose there is a dense subset of $f(M)$ where each fiber is connected, then each fiber of $f$ is ...
- 377
13
votes
3
answers
2k
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The "miracle" of Heegard Floer.
Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins ...
- 804
21
votes
5
answers
1k
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Explanation for E_8's torsion
To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is ...
- 15.1k
11
votes
3
answers
733
views
Relationship between universal coefficient theorem and $[K(\mathbb{Z},n), K(G,n)]$?
In short, I'm wondering whether the universal coefficient theorem can be understood/reinterpreted by using maps of Eilenberg-MacLane spaces. This is a wishy-washy idea and I don't have evidence to ...
- 6,069
4
votes
1
answer
1k
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properly interpreting Pi_0 in the homotopy exact sequence
Define the lens space L(m,n) as the quotient of S2m+1 by the action of the cyclic group ℤn⊂S1⊂ℂ*. We can create the infinite lens space L(∞,n) by a telescoping construction ...
- 6,069
184
votes
8
answers
12k
views
Two commuting mappings in the disk
Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk (or, if you prefer, the closed unit ball in $R^n$) to itself. Does there always exist a point $x$ such that $f(...
- 61.9k
14
votes
2
answers
984
views
Recovering a monoidal category from its category of monoids
What kind of additional properties and/or structures one needs to impose on the category
of (commutative or noncommutative) monoids of some monoidal category
so that one can recover the original ...
- 37.8k
9
votes
2
answers
699
views
Potential connected non-Lie subgroup
This painful question is inspired by the question
"non-Lie subgroups" . Let $f$ be a discontinuous additive map from $\mathbb{R} \to \mathbb{R}$. Is it possible that the graph of $f$, inside ...
- 156k
33
votes
6
answers
2k
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Is there a topology on growth rates of functions?
I've often idly wondered one can say about the collection of "growth rates". By growth rate, let's say we mean an equivalence class of functions $(0,\infty) \to (0,\infty)$, where two ...
- 793
3
votes
4
answers
627
views
Has anyone studied the applications which map open sets to either open or closed sets?
Consider two topological spaces X,Y and a function f from X to Y.
Are the following concepts already in use? How are they called?
1) f sends open subsets of X to either open or closed subsets of Y.
...
- 2,992
9
votes
5
answers
2k
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Analogues of the Weierstrass p function for higher genus compact Riemann surfaces
There was a previous post on the correspondence between Riemann surfaces and algebraic geometry. I want to ask a related but more detailed question.
BACKGROUND:
Engelbrekt gave an overview of how ...
- 3,968
9
votes
1
answer
2k
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What is Floer homology of a knot?
I've heard that there are different theories providing knot invariants in form of homologies. My understanding is that if you embed knot in a special way into a space, there is a special homology ...
- 15.1k
6
votes
2
answers
1k
views
Computations in Knot Homology Theories
1) Relative to one another, how computable are the various knot homology theories? For example, how many crossings can we allow a knot and still hope to compute its Khovanov homology, versus its knot ...
- 2,806
16
votes
6
answers
3k
views
Can any topological space be the result of a scheme?
Maybe this is trivial but lets give it a try anyways..
Obviously there is a forgetful functor from schemes to topological space.. but is it surjective on objects? i.e. I ask whether any topological ...
- 2,275