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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.

29
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7answers
4k views

Why is it useful to classify the vector bundles of a space?

It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces. Now, I've read somewhere that it is ...
7
votes
2answers
1k 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 ...
11
votes
5answers
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 ...
4
votes
1answer
329 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 ...
-4
votes
2answers
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 ...
2
votes
1answer
315 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 $\...
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 ...
6
votes
3answers
893 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 ...
13
votes
2answers
541 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 ...
6
votes
2answers
598 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} ...
4
votes
1answer
2k views

definition of the end of a manifold?

Hey everybody, I was hoping if somebody could help me out with the terminology. I've found that the "end of a manifold" is a function asigning to each compact set K a conected component e(K) of the ...
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 ...
2
votes
3answers
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): ...
17
votes
4answers
2k 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 ...
5
votes
1answer
765 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) =...
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. ...
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. $\...
-2
votes
2answers
789 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....
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
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 ...
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 ...
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\...
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 ...
26
votes
5answers
3k views

The ants-on-a-ball problem

Suppose I put an ant in a tiny racecar on every face of a soccer ball. Each ant then drives around the edges of her face counterclockwise. The goal is to prove that two of the ants will eventually ...
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 ...
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 ...
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?
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 ...
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 ...
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 ...
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?
13
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3answers
2k views

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 ...
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 ...
4
votes
1answer
778 views

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 ...
2
votes
2answers
186 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--->R^m ...
8
votes
2answers
778 views

Has anyone tabulated 2-knots? Would anyone like to try?

I'd love to have a list of 'small' 2-knots, for some sense of small. It's not clear what one should filter by, but there are two obvious candidates Write a movie presentation, and count the frames. ...
3
votes
4answers
505 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. ...
6
votes
2answers
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 ...
7
votes
1answer
2k views

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 ...
5
votes
2answers
363 views

Is a map that is locally fiberwise equivalent to a product a Hurewicz fibration?

The following is a result I feel like I've seen some form of before, but can't figure out how to prove or find a reference for. Suppose you have a map p:E \to B, with B paracompact, and suppose that ...
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4answers
3k views
1
vote
2answers
157 views

Something like Yoneda's lemma

This is inspired by The Whitehead for maps question. Consider two maps f, g: X\to Y which happen to induce the same maps (of discrete spaces) ...
1
vote
1answer
379 views

Equivalence of boundedness and total boundedness

Compact subspaces of metric spaces are totally bounded. In some spaces, however, this is equivalent to just being bounded. This (supposedly) holds in finite dimensional Banach spaces. Can we ...
5
votes
1answer
175 views

Homotopy type of stabilizers

Let X be a contractible metric space and G a topological group acting transitively on X (i.e. given any two points x,y \in X, there exists g \in G such that gx=y). My question is the following: is it ...
7
votes
1answer
469 views

Universal covers of domains in complex projective space

The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
9
votes
1answer
321 views

cardinality of final coalgebras in Top

Let P be a polynomial functor from Top to Top, by which I mean a functor of the form P(X) = ∐i ≥ 0 Si × Xi where the Si are finite sets, all but finitely many of which are empty. ...
8
votes
6answers
2k views

What is an example of a topological space that is not homotopy equivalent to a CW-complex?

It would also be nice if someone can explain this comment appearing on the Wikipedia page on CW-complexes: "The homotopy category of CW complexes is, in the opinion of some experts, the best if not ...
3
votes
2answers
368 views

Legendrian homotopy of curves in a contact structure?

I'm aware of the great body of work on Legendrian knot theory in contact geometry, but suppose I'm curious just about homotopy and not isotopy. How does one understand the space of Legendrian loops ...