# Questions tagged [gn.general-topology]

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

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### The Klee Trick for subsets of $\mathbb{R}^3$

Update: The lead paragraph has been changed to reflect the solution to a related question.
I asked the question Is dimension given by the Klee trick ever sharp?
and it has been answered in the ...

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320 views

### Zariski-homeomorphisms

This question is motivated by two questions at MO and
at MSE.
I am interested in homeomorphism types of (irreducible) complex-projective varieties with respect to the Zariski topology. Any two ...

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264 views

### Spaces that never separate the Hilbert cube

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.
Any finite dimensional space has this ...

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192 views

### How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...

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205 views

### Generalized Skorokhod spaces

Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...

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471 views

### Is there a natural topology on the set of open sets ?

Given a topological space $(X,\mathcal{O})$ can one assign a natural topology to $\mathcal{O}$ such that
1) The intersection of a compact set of open sets is again open,
2) The maps $\cap,\cup:\...

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390 views

### “Liftings” of L^\infty functions

This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...

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291 views

### The self-duality of topological compactness

The impatient reader can skip my attempt at motivation and go straight my "Question formulations for the impatient."
In a failed(?) attempt at discovering something new, some years ago I toyed with ...

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274 views

### In a locally contractible space can we find local bases of contractible sets whose closures are locally contractible?

In a locally contractible topological space $X$ is it possible at each point $x$ to find a local basis of contractible sets $U_i\ni x$ such that the closure of each set $\overline{U_i} \subset X$ is ...

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111 views

### A special connected subset of the Cantor fan

Is there a dense connected subset $X$ of the Cantor fan $$(C\times [0,1])/(C\times \{1\})$$ such that for every two connected subsets $X_1,X_2\subseteq X$, the intersection $X_1\cap X_2$ is connected?
...

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188 views

### Does anyone use non-sober topological spaces?

Recall that a sober space is a topological space such that every irreducible closed subset is the closure of exactly one point.
Is there any area of mathematics outside of general topology where non-...

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116 views

### Countable network vs countable Borel network

Definition. A family $\mathcal N$ of subsets of a topological space $X$ is called
$\bullet$ a network if for any open set $U\subset X$ and point $x\in U$ there exists a set $N\in\mathcal N$ ...

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61 views

### Is each Choquet topological group strong Choquet?

A topological space $X$ is called (strong) Choquet if the player II has a winning strategy in the (strong) Choquet game.
It is known that a metrizable space $X$ is
$\bullet$ Choquet if and only if ...

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99 views

### The automorphism group of the fibered cylinder

My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...

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96 views

### Spatiality of products of locally compact locales

In Johnstone´s Sketches of an Elephant Volume 2, page 716,
lemma 4.1.8 states that for spatial locales $X$ and $Y$ with $X$ locally compact then the locale product $X\times Y$ is spatial.
Is this ...

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90 views

### The first homotopic Baire class

Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ belongs to the first Baire class (to the first homotopic Baire class), if there exists a continuous map $H:X\times \omega\to Y$ (a continuous ...

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192 views

### Generalizing Gromov Hausdorff distance using Vietoris topology

There are two notions of convergence of a sequence of metric space. One is by the Gromov Hausdorff distance for compact metric spaces, another one is the pointed Gromov Hausdorff convergence for ...

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147 views

### The Fubini property of the $\sigma$-ideal generated by closed subsets of measure zero

Question. Can a Borel subset $B\subset\mathbb R\times \mathbb R$ be covered by countably many closed sets of measure zero in the plane if for every $(a,b)\in\mathbb R\times\mathbb R$ the horisontal ...

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136 views

### Quotients of 4-sphere by smooth $Z_p$ actions with knotted fixed point sets

This question is closely related to another I asked today.
Giffen showed in 1966 that the generalized Smith conjecture is false by constructing for odd $p$ a smooth $Z_p$ action on $S^4$ with fixed-...

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241 views

### Topologically transitive, pointwise minimal systems

I'm cross-posting this from SE.
Let $T$ be a group, and let $(X,T)$ be a flow, i.e. $X$ is a compact Hausdorff space and $T$ acts on $X$ by homeomorphisms. A flow $X$ is called topologically ...

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176 views

### Are there any known ``topological" invariants for branched coverings?

My question is the following: let $f:\Omega\to \mathbb{R}^n$ be a branched covering, namely $f$ is continuous, discrete (each fiber is a discrete subset of $\Omega$) and open (open sets get mapped ...

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101 views

### topological spaces admitting CAT(1) metrics

Suppose that $X$ is a locally contractible completely metrizable topological space. Is it true that $X$ can be metrized as a (complete) CAT(1) metric space?
The only result in this direction I know ...

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100 views

### Large discrete subspaces in spaces of separately continuous functions

For topological spaces $X,Y,Z$ let $SC_p(X\times Y,Z)$ be the space of separately continuous functions $f:X\times Y\to Z$ endowed with the topology of pointwise convergence.
It is easy to see that ...

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224 views

### A compactification of the non-negative rationals with the discrete topology

Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is ...

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533 views

### Homeomorphisms of product spaces: an example

In the first of these lectures (http://www.mpim-bonn.mpg.de/node/4436) given by M. Freedman he says that there exists (compact metric) spaces $X$ and $Y$ such that $X\times S^{1}$ is homeomorphic to $...

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176 views

### Tensor product of dual groups

Let $G,H$ be compact abelian groups, $G^*,H^*$ be their Pontryagin duals, $G^*\otimes H^*$ the tensor product of $G^*,H^*$ and $K=(G^*\otimes H^*)^*$. Does the group $K$ have a special name? What is ...

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540 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 ...

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3k views

### Convex hulls of compact sets

Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?

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492 views

### Is the Milnor construction contractible

Let $G$ be any topological group. Then we can form the infinite join $E_G$ of $G$, i.e. the colimit $G*G*G\cdots$.
Is $E_G$ contractible?
I mean it is clear that $E_G$ is weakly contractible, but ...

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146 views

### Pettis Integrability and Laws of Large Numbers

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...

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212 views

### Series in topological rings that only converge if almost all summands are zero

While trying to understand a certain topological ring better, I stumbled onto the following question.
Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a ...

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### Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure.
Notation used ...

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### How to call a point in a space having the property that there is essentially one $\omega$-sequence converging to it?

Consider the point $x=\langle \omega_1,\omega\rangle$ in the Tychonov plank $(\omega_1 + 1)\times(\omega + 1)$. Then there is essentially only one sequence (of length $\omega$) converging to it, ...

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306 views

### Terminology for notion dual to “support”

If $X$ is a set (feel free to think of it as finite, but it doesn't have to be) and $f$ a real function on $X$, call the support $\operatorname{supp} f$ the subset of $X$ consisting of all elements $i\...

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437 views

### Closure properties of familes of $G_\delta$ sets.

Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space ...

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412 views

### Does local strict contractibility imply ANR?

Say that a space (= compact metrizable space) $X$ is locally strictly contractible if, for every $p\in X$ and neighborhood $U$ of $p$, there is a neighborhood $V$ of $p$ which can be contracted to $p$ ...

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846 views

### What relates to measure spaces as topological spaces relate to metric spaces ?

Has there been study of a generalization of measure spaces along the following or similar lines ?
Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball of radius $r$ of $U$ ...

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506 views

### Continuous images of Cantor cubes

The original title of this question was "Is there only one (up to homeomorphism) zero-dimensional homogeneous dyadic space of weight $\mathfrak{c}$?". I changed it with the hope of getting a bit more ...

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266 views

### How to use the Lefschetz trace formula on infinite dimensional spaces?

I think the Lefschetz trace formula says something like:
if $F: X \to X$ is a continuous map of compact manifolds, then
$\chi(X^F) = \sum (-1)^i \mathrm{Tr} f_*|_{H_i(X)}$
First of all, this ...

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331 views

### The Space of Cellular Maps

Let $X$ and $Y$ be CW complexes.
Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...

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409 views

### Ever seen a ringed group?

A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...

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443 views

### The Mapping Cylinder of a Pullback Square

Suppose I have a pullback square, which I think of as a map from the fibration
$q:X\to A$ to the fibration $p:Y\to B$. Then there is an induced map $m: M \to N$
from the mapping cylinder $M$ of $X\...

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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 ...

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116 views

### For which topological spaces does pullback along $\operatorname{ev}_0:B^I\to B$ have a right adjoint?

Let $B$ be a topological space. Consider the evaluation at zero of paths in $B$. This is a continuous map $\operatorname{ev}_0:B^I\to B$ where the domain carries the compact-open topology.
For which ...

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84 views

### Is each Peano continuum a topological fractal?

Problem. Is each Peano continuum a topological fractal?
A compact Hausdorff space $X$ is a topological fractal if $X=\bigcup_{i=1}^n f_i(X)$ for some continuous maps $f_1,\dots,f_n:X\to X$ such that ...

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237 views

### Grothendieck letter to Jun-Ichi Yamashita on tame topology

I am looking for Grothendieck writings on tame topology:
a manuscript on tame topology mentioned by Scharlau; a letter to Jun-Ichi Yamashita; a letter to Z.Mebkhout.
I am also interested in ...

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### An easier example of complete lattice such that the Scott topology on it is not sober

Basic notions: $1$, A partially ordered set is a dcpo if each of its directed subsets has a supremum. (https://en.wikipedia.org/wiki/Complete_partial_order)\
$2$, A subset O of a dcpo P is called ...

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212 views

### The second dual of $C(X)$ with the compact-open topology

Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...

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### Local isometry of complete length spaces that is not a covering map

Let $\pi:\widetilde{M}\to M$ be a surjective local isometry between complete length spaces (local isometry means that every point $x\in \widetilde{M}$ has a neighborhood which is isometrically mapped ...

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149 views

### On generically Haar-null sets in the real line

First some definitions.
For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...