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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.

12
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6answers
3k views

A good place to read about uniform spaces

I'd like to learn a bit about uniform spaces, why are they useful, how do they arise, what do they generalize, etc., without getting away from the context of general topology. I have to prepare an ...
32
votes
1answer
845 views

Is $L^2(\mathbb R)$ homeomorphic to $L^1(\mathbb R)$?

Is $L^2(\mathbb R)$ homeomorphic to $L^1(\mathbb R)$? More generally, are there instances of surprising homeomorphisms between non-isomorphic Banach spaces?
30
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3answers
1k views

Can a connected planar compactum minus a point be totally disconnected?

What the title said. In a slightly more leisurely fashion:- Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x\}$ be ...
28
votes
2answers
2k views

Is $\mathbb{C}^2$ homeomorphic to $\mathbb{C}^2 - (0,0)$ with the Zariski topology?

A fellow grad student asked me this, I have been playing for a while but have not come up with anything. Note that $\mathbb{C}$ is homeomorphic to $\mathbb{C} - \{0\}$ in the Zariski topology - just ...
23
votes
3answers
661 views

What sets of self-maps are the continuous self-maps under some topology?

An open question on MSE, https://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a ...
18
votes
2answers
1k views

Rugged manifold

It is well known that any compact smooth $m$-manifold can be obtained from $m$-ball by gluing some points on the boundary. Is it still true for topological manifold? Comments: To proof the smooth ...
16
votes
5answers
6k views

Locally compact Hausdorff space that is not normal

What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of Urysohn'...
15
votes
1answer
880 views

Pullbacks as manifolds versus ones as topological spaces

My question is: Does the forgetful functor F:(Mfd) $\to$ (Top) preserve pullbacks? Detailed explanation is following. A pullback is defined as a manifold/topological space satisfying a universal ...
15
votes
1answer
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 ...
28
votes
2answers
1k views

Does there exist any non-contractible manifold with fixed point property?

Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the ...
21
votes
8answers
1k views

Self-containing structures

(This question is partly inspired by What is inter-universal geometry?.) I have absolutely no background in Teichmuller theory or any related subject, but what I can follow of Mochizuki's description ...
8
votes
2answers
1k views

Is there a conceptual reason why topological spaces have quotient structures while metric spaces don't?

Of the mathematical objects that I am familiar with, it is normally the case that the product of 2 objects is an object of the same type and that an equivalence relation on an object induces a ...
5
votes
2answers
3k views

On the cohomology of a finite covering map

So let $X$ be a "nice" topological space and assume that $G$ is a finite group which acts freely on $X$. Q: Is there a simple relationship between the cohomology groups $H^i(G,\mathbf{Z}), H^i(X,\...
27
votes
1answer
1k views

How many polynomial Morse functions on the sphere?

Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function. If $f$ is a Morse function of degree $1$, you ...
21
votes
3answers
1k views

An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request

There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter. But there's ...
20
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0answers
649 views

Are amenable groups topologizable?

I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is ...
18
votes
3answers
1k views

Non-homeomorphic spaces such that taking away a point makes them homeomorphic

Are there topological spaces $X,Y$, each having more than $2$ points, satisfying the following two properties? $X\not\cong Y$, and there is a bijection $\varphi: X\to Y$ such that for all $x\in X$ ...
18
votes
1answer
663 views

Closed totally disconnected subspaces

It is a remarkable property of uncountable compact metric spaces that each of them contains a homeomorphic copy of the Cantor set. In general, one cannot expect containment of Cantor cubes (in ...
18
votes
1answer
660 views

Can two-point sets be Borel?

Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914. I wonder if the following question of ...
18
votes
4answers
2k views

Compact open topology on $\mathrm{Homeo}(X)$

Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)...
15
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1answer
1k views

Foundations of topology

I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here. Also some time ago I read ...
9
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3answers
2k views

Where can I find a proof of the de Rham-Weil theorem?

Where can I find a proof of the de Rham-Weil theorem? Does anyone know?
9
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1answer
502 views

Topology from the viewpoint of the filter endofunctor

Question. Are there any references that develop general topology from the viewpoint of a functor $$\Phi : \mathbf{Rel} \rightarrow \mathbf{Rel}$$ that assigns to every set $X$ the set $\Phi(X)$ of ...
18
votes
1answer
625 views

convexity of images of space-filling curves

Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t \...
17
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2answers
1k views

compact-open topology on $B(H)$

In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces. Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...
15
votes
0answers
974 views

Connected and locally connected, but not path-connected

Allow me to use some non-standard terminology: A h-contractible space is a non-empty topological space $X$ such that, for any topological space $T$ and any pair of continuous maps $f_0, f_1 : T \to X$...
14
votes
1answer
568 views

Bijection $f: \mathbb R^n \to \mathbb R^n$ that maps connected onto connected sets must map closed connected onto closed connected sets?

Willie Wong asked here (MO) and here (MSE) very interesting question. As he phrased it: Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$...
14
votes
2answers
1k views

Generalization of Darboux's Theorem

Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$. Does any of the following generalizations Let $U\...
14
votes
3answers
504 views

Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?

This question is related to another one that I asked two days ago. Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with the following two properties? The ...
13
votes
3answers
970 views

What spaces can be obtained from $\mathbb{R}^{n}$ by taking quotient spaces and subspaces?

Is there a good characterization of the smallest collection of topological spaces which contains $\mathbb{R}^{n}$ for each $n$, and is closed under taking subspaces and quotient spaces? A bit of ...
12
votes
1answer
600 views

Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?

It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In ...
6
votes
2answers
759 views

Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question. I can think of at least four topologies to put on $C_c(M)$: Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney $C^\...
20
votes
2answers
665 views

The Gelfand duality for pro-$C^*$-algebras

The Gelfand duality says that $$X\to C(X)$$ is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
19
votes
0answers
427 views

Disc bounded by a plane curve

Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$. Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve? It is easy to find an open disc ...
15
votes
5answers
2k views

Compactness of the Hilbert cube without the Axiom of Choice

I am just curious: is there a published proof of the compactness of the Hilbert cube that does not use the Axiom of Choice, or is it well known?
13
votes
1answer
1k views

Characterization of Stone-Cech compactifications

Suppose I have an infinite discrete topological space $X$ of cardinality $\kappa$. Then I know some things about the Stone-Cech compactification, $\beta X$: it is Hausdorff and compact but not ...
13
votes
1answer
393 views

Continuum Hypothesis and the fact that every co-finite topological space, with uncountable underlying set , is contractible

Let $X$ be a co-finite topological space. If $|X| \ge 2^{\aleph_0}=\mathfrak c$, then $X$ is contractible (https://en.wikipedia.org/wiki/Contractible_space) . Indeed, there is a bijection $f: X \times ...
13
votes
2answers
1k views

Well-pointed space which is not locally contractible

I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the ...
13
votes
1answer
717 views

Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
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 ...
11
votes
0answers
159 views

Shift invariant measurable selection theorem

Let $(X,\mathcal{F})$ be some measure space and endow $\mathbb{R}^\mathbb{Z}$ with the product topology and borel $\sigma$-field. Let $F$ be a point to set mapping $X^\mathbb{Z}\rightarrow \mathcal{P}(...
10
votes
0answers
512 views

A standard name for a function satisfying the intermediate value theorem?

Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property: $(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
9
votes
1answer
343 views

Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre. Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
9
votes
1answer
491 views

Existence of infinite groups that are too reluctant to be topological

With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...
9
votes
2answers
2k views

Quotient of metric spaces

Let $(X,d)$ be a compact metric space and $\sim$ an equivalence relation on $X$ such that the quotient space $X/\sim$ is Hausdorff. It is well known that in this case the quotient is metrizable. My ...
9
votes
4answers
1k views

Largest Hausdorff quotient

The inclusion of the full subcategory of Hausdorff topological spaces into the category of topological spaces has a left adjoint, which can be proven easily by the Adjoint Functor Theorem (see for ...
7
votes
2answers
347 views

Refining open covers in locally path connected spaces

Suppose $X$ is a locally path connected topological space and $\mathcal{U}$ is an open cover of $X$ (consisting of path connected sets if we want). One often wants the intersection $A\cap B$ of ...
7
votes
2answers
1k views

End point compactification for metric spaces

Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here). For ...
6
votes
0answers
541 views

Intersections of open sets and $\alpha$-favorable spaces

I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the ...
6
votes
2answers
394 views

Hausdorff group topologies on finitely generated groups

Suppose $G$ is a finitely generated Hausdorff topological group. Must $G$ be first countable (or perhaps a sequential space)? What if we restrict to the abelian case? I wonder if this is even true ...