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Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

3
votes
3answers
679 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 ...
12
votes
4answers
975 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 ...
9
votes
2answers
818 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 ...
38
votes
4answers
3k 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 ...
28
votes
4answers
4k 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 ...
4
votes
3answers
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 ...
5
votes
2answers
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 ...
6
votes
1answer
229 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, ...
11
votes
6answers
2k 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. ...
32
votes
14answers
4k 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 ...
-4
votes
1answer
5k 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 ...
22
votes
2answers
895 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 ...
5
votes
1answer
764 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) =...
6
votes
10answers
3k 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). ...
35
votes
4answers
3k 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 ...
8
votes
1answer
517 views

Why is Top_4 a reflective subcategory of Top_3?

Hi, 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 ...
34
votes
3answers
4k 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. $\...
1
vote
1answer
160 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 $S$ ...
-2
votes
2answers
788 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 ...
-4
votes
4answers
618 views

What is the max number of points in R^3, interconnected by generic curves?

The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of it....
1
vote
10answers
5k views

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 ...
2
votes
3answers
763 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 is ...
7
votes
2answers
3k 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 ...
2
votes
1answer
436 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 ...
18
votes
15answers
14k 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
2answers
477 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
3answers
534 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 ...
12
votes
3answers
703 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 ...
12
votes
2answers
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 ?
20
votes
4answers
1k 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. ...
7
votes
1answer
515 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\...
6
votes
2answers
36k 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 ...
3
votes
1answer
296 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 ...
1
vote
2answers
639 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?
3
votes
1answer
1k views

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 ...
15
votes
2answers
879 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 ...
2
votes
3answers
347 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 ...
35
votes
8answers
3k views

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 ...
4
votes
1answer
427 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 ...
4
votes
1answer
304 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 ...
52
votes
13answers
3k 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 ...
3
votes
1answer
212 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 ...
41
votes
8answers
5k 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 ...
12
votes
3answers
416 views

Groupoid of moves on trivalent fatgraph

Let $T$ be a finite trivalent fatgraph - i.e. a graph with a cyclic order of the edges at each vertex. Then there are certain basic "moves" we can perform on $T$: an embedded edge can be collapsed and ...
5
votes
6answers
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 ...
23
votes
8answers
3k 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) ...
3
votes
4answers
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?
9
votes
1answer
423 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 ...
11
votes
3answers
751 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 ...
2
votes
1answer
246 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 ...