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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
0answers
245 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 ...
11
votes
2answers
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 ...
6
votes
0answers
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 ...
7
votes
2answers
770 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 ...
14
votes
5answers
621 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 fit the ...
2
votes
1answer
450 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 ...
5
votes
2answers
618 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 ...
32
votes
4answers
3k views

Reference for the Gelfand-Neumark theorem for commutative von Neumann algebras

The Gelfand-Neumark 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 ...
3
votes
1answer
358 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 $...
12
votes
2answers
540 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 ...
5
votes
2answers
898 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 complete metric ...
8
votes
4answers
2k 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
3answers
993 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 ...
20
votes
7answers
8k 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 ...
-4
votes
1answer
451 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 ...
3
votes
1answer
636 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 $...
7
votes
3answers
832 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
votes
2answers
528 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 ...
5
votes
1answer
391 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, ...
1
vote
3answers
840 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 ...
33
votes
2answers
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 ...
2
votes
0answers
603 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?
2
votes
5answers
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 ...
3
votes
2answers
927 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 ...
40
votes
3answers
7k 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 $\phi_*[N]\in H_*(M)$ ...
14
votes
5answers
2k 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 "$...
3
votes
3answers
399 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 ...
7
votes
4answers
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)?
6
votes
1answer
283 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
vote
1answer
91 views

Approximate selection theorems for factoring through perfect maps

I have the following setup: $X, Y$ are topological spaces (if it helps, they can both be $T_1$ and normal. They can even be countably paracompact. They can't be assumed paracompact). $V$ is a normed ...
5
votes
0answers
534 views

continuous selection of a multivalued function?

The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...
3
votes
1answer
339 views

Is ω1 × βN normal?

Once upon a time I asked whether $\omega_1 \times \beta \mathbb{N}$ is normal. I got the answer no and a fairly convincing proof of this here However I'm currently in a situation where I have three ...
5
votes
2answers
610 views

$2^{\omega_1}$ separable?

I was rereading an answer to an old question of mine and it included a reference to the fact that $2^{\omega_1}$ was separable. I'm having a hard time finding a reference for this fact, and the proof ...
6
votes
1answer
394 views

Homomorphisms of Topological Groups which are Automatically Fiber Bundles?

Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will ...
24
votes
6answers
2k views

Applications of string topology structure

Chas and Sullivan constructed in 1999 a Batalin-Vilkovisky algebra structure on the shifted homology of the loop space of a manifold: $\mathbb{H}_*(LM) := H_{*+d}(LM;\mathbb{Q})$. This structure ...
3
votes
2answers
1k views

Paracompact but not Hausdorff

Do paracompact non-Hausdorff spaces admit partions of unity? I'm just curious.
21
votes
3answers
2k views

The Closure-Complement-Intersection Problem

Background Let $A$ be a subset of a topological space $X$. An old problem asks, by applying various combinations of closure and complement operations, how many distinct subsets of $X$ can you ...
12
votes
4answers
925 views

Topologizing free abelian groups

For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
6
votes
1answer
403 views

Countable paracompactness, normality and locally countable open covers

(repost from the topology Q&A board) I have a (T_1), Normal, countably paracompact space X. I would like to know if every locally countable open cover of X (i.e. an open cover such that every x ...
6
votes
2answers
461 views

Can I detect the point of impact without looking at it?

I'm going to postpone the motivation for this question because the question itself involves no complicated maths and may well have a very simple solution so I don't want to put anyone off with high ...
1
vote
1answer
575 views

When is the realization of a simplicial space compact ?

Suppose $X$ is a simplicial space of dimension $M$ (i.e. all simplices above dimension $M$ are degenerate). The claim is: $|X|$ is compact. iff $X_n$ is compact for each $n$. Suppose each $X_n$ is ...
3
votes
2answers
454 views

Euler characteristics and operator indices as exponents for Laurent polynomials

This question is rather vague. Are there any natural situations which involve Laurent polynomials of the form $$\sum q^{a_i}\in\mathbb{Z}[q,q^{-1}]$$ where the $a_i$'s are either Euler characteristics ...
4
votes
0answers
278 views

What is enough to conclude that something is a CW complex (part II)?

A while ago I asked a question about recoqnizing CW complexes and got an extremely nice and concrete answer. However, I am still interested in a more general treatment of this and therefore pose the ...
-3
votes
2answers
300 views

Dispensing with the notion of infinity for the sake of coverings [closed]

Instead of taking a one to one correspondence meaning each set has the same number of elements. why not use the concept of coverings of topology? The irrational numbers covers the whole numbers but ...
23
votes
2answers
3k views

How do you axiomatize topology via nets?

Let $X$ be a set and let ${\mathcal N}$ be a collection of nets on $X.$ I've been told by several different people that ${\mathcal N}$ is the collection of convergent nets on $X$ with respect to some ...
24
votes
17answers
9k views

Applications of Brouwer's fixed point theorem

I'm presenting Brouwer's fixed point theorem to an audience that knows some point-set topology. Does anyone have any zippy / enlightening / cool applications or consequences of it? So far, I have: ...
5
votes
7answers
3k views

Do the empty set AND the entire set really need to be open? [closed]

My question is motivated by the previous discussion 'Why is a topology made of open sets?'. While the axioms for arbitrary unions and finite intersections are without doubt essential to the concept of ...
2
votes
0answers
201 views

Is the realization of a proper map of simplicial spaces proper ?

Let $f:X \rightarrow Y$ be a map of $m$-dimensional simplicial spaces (which means that all simplices above dimension $m$ are degenerate). Recall, that $f$ is a natural transformation of functors from ...