# Questions tagged [gn.general-topology]

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

4,379
questions

0
votes

0
answers

135
views

### Define differentiability in a quotient space that is NOT a manifold but close to an Euclidean space

Suppose we have $X = R^n$, fixed $x_0 \in R^n$ and a polynomial function $f$ by which we define a set $C = \{x \in R^n \mid f(x) = x_0\}$. Consider the quotient space $X/C$, assume $C$ is closed and $...

3
votes

1
answer

107
views

### Existence of Borel uniformization for coanalytic set with non-$K_\sigma$ sections

Suppose that $X$ is a Polish (or standard Borel) space and $\omega^\omega$ is the Baire space of all natural number sequences. My question is: If $A\subseteq X\times \omega^\omega$ is a coanalytic set ...

0
votes

0
answers

73
views

### Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?

(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?)
Assume $(\Omega, \mu)$ is a probability space. Consider a ...

2
votes

1
answer

199
views

### Is the Fortissimo space on discrete $\omega_1$ radial?

Let $\omega_1$ have the discrete topology. Its Fortissimo space is $X=\omega_1\cup\{\infty\}$ where neighborhoods of $\infty$ are co-countable.
A space is radial provided for every subset $A$ and ...

1
vote

0
answers

62
views

### Construct manifold given simplical complex

It's known that, in general, given a simplical complex, answering if it's homeomorphic to a manifold is undecidable. However, given a positive answer to the problem, is there an algorithm to construct ...

2
votes

1
answer

75
views

### Mandelbrot boundary and component of $\infty$

Let $M$ be the Mandelbrot set, and $\partial M$ its boundary. So $\partial M$ is the set of those points $z\in M$ such that every neighborhood of $z$ contains a point of $\mathbb R^2\setminus M$.
Let $...

1
vote

0
answers

67
views

### Approximating evalutation maps at open sets over invariant measures

Let $G$ be a group acting by homeomorphisms on a compact metrizable space, say $X$; let's denote by $\alpha:G\to\mathrm{Homeo}(X)$ the action, $g\mapsto\alpha_g$, and consider the weak-$^*$ compact ...

1
vote

0
answers

52
views

### Continuous maps between compact space and cubes

Let $X$ be a compact metrizable space. Let $f$ be a continuous map from $X$ to the cube $[0,1]^m$. I would like to know under which condition of a continuous map $g: X\to [0,1]^n$ there exists a ...

4
votes

1
answer

153
views

### Compact-open Topology for Partial Maps?

I asked the same question on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow.
Compact open topology is one of the most common ways of ...

3
votes

0
answers

94
views

### Metrizing pointwise convergence of *sequences* of functionals in a dual space

This question was asked by myself on the math stackexchange a few days ago. I thought I'd repeat it here:
Let $X$ be a normed, real vector space of uncountable dimension. Let $X^*$ denote the set of ...

2
votes

1
answer

173
views

### Simple closed curves in a simply connected domain

Let $U$ be a bounded simply connected domain in the plane. Let $K$ be the boundary (or frontier) of $U$. For every $\varepsilon>0$ is there a simple closed curve $S\subset U$ such that the ...

0
votes

0
answers

21
views

### How exactly does one construct a covering space corresponding to a subgroup [migrated]

I'm trying to understand how to construct covers based on Hatcher and I'm using this question to understand it.
Let $B = B({a, b})$ be the wedge of circles.
$F(a, b)$ a free group on $\{a, b\}$.
Let $...

3
votes

0
answers

85
views

### Progess on conjectures of Palis

I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of ...

0
votes

1
answer

50
views

### Continuous selectors of a continuous multifunctin on a compact metric space

I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector.
...

9
votes

1
answer

501
views

### Are “most” bounded derivatives not Riemann integrable?

Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...

2
votes

2
answers

309
views

### Is a Hausdorff separable topological space that is uniform and complete necessarily a Polish space?

Is an Hausdorff separable topological space that is uniform and complete necessarily a Polish space ?

4
votes

1
answer

195
views

### Why is this continuum circle-like?

A continuum is a compact connected metrizable space.
A continuum $X$ is called arc-like if for every $\varepsilon>0$ there is an open cover $U_1,\ldots,U_n$ of $X$ such that the diameter of $U_i$ ...

5
votes

5
answers

871
views

### Two arcs in the complement of a disc must intersect?

Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$.
Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...

6
votes

4
answers

473
views

### Countable chain condition in topology

A topological space $X$ is said to have the countable chain condition (ccc) if every collection of open and disjoint subsets of $X$ is at most countable. This definition can be found in L. Steen, J. ...

2
votes

1
answer

227
views

### Hahn-Banach theorem and ultrafilter lemma

I'm unable to understand a remark in "Two application of the method of construction by ultrapowers to analysis" by Luxemburg, which uses the ultrafilter lemma to prove the Hahn-Banach ...

2
votes

2
answers

141
views

### $String/CP^{\infty}=Spin$ or a correction to this quotient group relation

We know that there is a fiber sequence:
$$
... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ...
$$
Is this fiber sequence induced from a short exact sequence?
If so, is that
$$
1 \to B^2 Z = B S^...

1
vote

0
answers

122
views

### Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?

What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
Namely, how do we know
$$
K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)?
$$
Naively -- in each step ...

1
vote

0
answers

201
views

### Examples of when $X$ is homotopy equivalent to $X\times X$

I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....

0
votes

0
answers

152
views

### Are all infinite-dimensional Lie groups noncompact?

Basically what the title says — if a Lie group is infinite-dimensional, is it necessarily noncompact?

8
votes

0
answers

244
views

### Is the Mandelbrot set Suslinian?

The Mandelbrot set is known to be (path-)connected and compact. A non-degenerate space with these properties is called a continuum.
A continuum $X$ is Suslinian if every collection of non-degenerate ...

2
votes

0
answers

86
views

### Explicit CW-complex replacement of the space of reparametrization maps

Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...

10
votes

1
answer

351
views

### Two dimensional perfect sets

Consider the following family of sets
$$ \begin{align*}
\mathcal{F} = \{X\subseteq [0,1]\times [0,1] \mid \ &X \text{ is closed and }\\& \forall x \in \pi_0 (X) (\{y \in [0,1] \mid (x,y) \in ...

0
votes

2
answers

291
views

### If a graph embedded on a surface is divided by a curve into a right and left that do not intersect can it be embedded on a surface of smaller genus?

Suppose we have a graph $G$ embedded on a (smooth, orientable etc) surface $Q$. Suppose there is a cycle $C$ of $G$ such that
$C$ does not separate our surface $Q$ into two connected regions and ...

1
vote

1
answer

153
views

### Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?

Because flowmaps are homeomorphic maps, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain dynamical system?
that is, does ...

1
vote

0
answers

117
views

### Can a closed null-homotopic curve be filled in by a disc?

Let $U\subseteq\Bbb R^n$ be an open set and $\gamma\subset U$ a closed null-homotopic curve in $U$ (i.e. it can be contracted to a point). Then is there an embedded disc $D\subset U$ with boundary $\...

2
votes

0
answers

93
views

### Unordered configuration space with non-distinct points

Consider a topological space $X$, a natural number $n>0$ and
the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if
there is a permutation $\sigma$ ...

7
votes

1
answer

234
views

### Does the continuous image of a disc contain an embedded disc?

Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...

3
votes

1
answer

142
views

### Approximating continuous functions from $K\times L$ into $[0,1]$

Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...

10
votes

0
answers

144
views

### Can the nowhere dense sets be more complicated than the meager sets?

Suppose $X$ is a completely metrizable space with no isolated points. Let $\mathcal{ND}_X$ denote the ideal of nowhere dense subsets of $X$, and let $\mathcal{M}_X$ denote the ideal of meager subsets ...

1
vote

1
answer

67
views

### Is this notion of being "fully" convex closed under set addition?

While reading through "Linear Operators: General theory" by "Jacob T. Schwartz", reading the corollary to II.10.1 which states that for a compact convex subset $C$ of some ...

6
votes

0
answers

204
views

### Generalizing uniform structures as Grothendieck topologies

Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...

8
votes

0
answers

201
views

### A variation of necklace splitting

Our problem is the following:
Let $n$ and $k$ be integers.
We are given two (unclasped) necklaces, each with $n$ colored stones: a top necklace which has $k$ colors and a bottom necklace which has 2 ...

0
votes

0
answers

116
views

### Uncountable collections of sets with positive measures

Let $X$ be a compact metric space and let $T: X \rightarrow X$ be continuous. Let $\mu$ be a $T$-invariant Borel probability measure (which we can always find by the Krylov-Bogoliubov theorem).
Let $(...

13
votes

2
answers

660
views

### Smooth Urysohn's lemma on Fréchet spaces

Let $V$ be a Fréchet topological vector space.
Let $K_0$ and $K_1$ be two closed subsets which are disjoint.
I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$
whose restriction ...

4
votes

0
answers

156
views

### When $X$ is homeomorphic to $\mathscr{F}[X]$?

While I was talking to some colleagues, one of them said that there exists a topological space $X$ such that $X$ is uncountable, non-discrete and homeomorphic to $\mathscr{F}[X]$ (the Pixley-Roy ...

8
votes

1
answer

328
views

### "Compactness length" of Baire space

Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton?
In more ...

11
votes

1
answer

309
views

### Density of linear subspaces in $C(K)$

Let $K$ be a compact Hausdorff space and denote by $C(K)$ the space of all real valued and continuous functions on $K$. We endow $C(K)$ with the supremum norm topology, making it a Banach space.
...

7
votes

0
answers

143
views

### The space of analytic associative operations

This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...

4
votes

3
answers

548
views

### Does there exist a topological space $X$ such that $X^2$ and $[0,1]$ are homeomorphic?

I have proved that if $X$ is not connected then $X^2$ is not connected either. So my idea was to prove that if $X$ is connected then $X^2$ blown up any point is also connected. But I don't know ...

4
votes

0
answers

138
views

### Two other variants of Arhangel'skii's Problem

This question is a follow up to another question of mine, which turned out to be easy (for background on Arhangel'skii's Problem see Arhangel'skii's problem revisited). Recall that a space is ...

2
votes

1
answer

180
views

### Is a simple closed curve always a free boundary arc?

Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points?
For a simple closed curve $\...

15
votes

3
answers

1k
views

### Is symmetric power of a manifold a manifold?

A Hausdorff, second-countable space $M$ is called a topological manifold if $M$ is locally Euclidean. Let $SP^{n}(M): = \left(M \times M \times \cdots \times M \right)/ \Sigma_{m}$, where product is ...

2
votes

0
answers

109
views

### Homotopy type of a 3-manifold produced via Dehn surgery?

My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology.
I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ ...

0
votes

1
answer

152
views

### Distinguishable under manifold topology but indistinguishable under the Alexandrov topology

Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal.
In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...

6
votes

0
answers

170
views

### Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?

I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...