# Questions tagged [gn.general-topology]

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

**31**

votes

**3**answers

942 views

### Non embedding of $Y\times Y$ into $\mathbb{R}^3$

I know that this is a well known result, but where can I find a proof? I am also interested to see more general non-embedding results of this type.
Theorem. Let $Y$ be the union of two segments ...

**2**

votes

**1**answer

50 views

### Is each cometrizable space a subspace of a cometrizable topological group?

Following Gruenhage we call a topological space $X$ cometrizable if $X$ admits a weaker metrizable topology such that every point $x\in X$ has a (not necessarily open) neighborhood base consisting of ...

**3**

votes

**1**answer

93 views

### Prove that $\mu \left(\left\{t\in X\,;\;\sum_{i=1}^d|\phi_i(t)|^2>r \right\}\right)=0$

Let $(X,\mu)$ be a measure space and $\phi=(\phi_1,\cdots,\phi_d)\in L^{\infty}(X)$.
Let
$$r=\max\left\{\sum_{i=1}^d|z_i|^2; (z_1,\cdots,z_d)\in \mathcal{C}(\phi)\right\},$$
where $\mathcal{C}(\phi)$...

**3**

votes

**1**answer

94 views

### Hausdorff measure of intersection of a ball and a set in $\mathbb {R} ^ n$

Let $A$ a subset of $\mathbb R ^n$, $B=B(x,r) \subset \mathbb {R} ^n$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $\mathbb R ^n$ by $\mathcal H^{n-1}$. Also assume that $\...

**2**

votes

**0**answers

99 views

### Fiber-bundle : continuity of transition maps and inverse in general

Let $(E,\pi,B)$ be a locally trivial fibration, with fiber a topological space $F$, $\Phi_i$ and $\Phi_j$ two trivializations over $U_i$ and $U_j$. The transition map from $i$ to $j$ is the ...

**2**

votes

**1**answer

94 views

### Irreducible subcontinuum of Lorenz attractor?

In my first question Lorenz attractor path-connected?, some are saying the Lorenz attractor $\mathscr L$ is not path-connected.
But suppose $x$ and $y$ are two points in different path components of ...

**2**

votes

**0**answers

122 views

### 3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators

During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...

**5**

votes

**1**answer

152 views

### Proper homotopy

Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper?
I am interested in particular in ...

**2**

votes

**1**answer

690 views

### A new generalisation of dimension? part 2

I worked this theory : A new generalization of the dimension?
I have a theorem about dimensions which is more general and simple than for matroids.
Definition 1: A structure $S$, is a pair $(X, \...

**9**

votes

**1**answer

511 views

### Lorenz attractor path-connected?

Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure.
EDIT: The answer below is unsatisfactory, and possibly ...

**15**

votes

**1**answer

407 views

### The dominating number $\mathfrak{d}$ and convergent sequences

All spaces considered below are compact Hausdorff.
If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...

**2**

votes

**0**answers

110 views

### What is the boundary of the set $\{ x : dist (x ,\partial \Omega) > \alpha \}$ for a domain $\Omega$?

Let $\Omega$ is a bounded open domain in $\mathbb R ^n$, and $\alpha \geq 0$ a real number, and consider the set $ E_\alpha = \{ x \in \Omega : \text{dist}(x , \partial \Omega) > \alpha\} $, which ...

**6**

votes

**1**answer

125 views

### Objects whose morphisms are Lipschitz maps

I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz").
The answer seems pretty clear, and proceeds like the definition of manifolds:
1) If $...

**6**

votes

**2**answers

314 views

### What is the name for a set endowed with a Lipschitz structure?

I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the ...

**6**

votes

**1**answer

113 views

### Example similar to the Griffiths twin cone but with fundamental group that allows surjection onto $\mathbb Z$

The Griffiths twin cone is an example of a wedge sum of two contractible spaces being non-contractible. Namely, it is the wedge sum $\mathbb G=C\mathbb H\vee_p C\mathbb H$ of two coni over the ...

**9**

votes

**1**answer

187 views

### What is known about topological groups of countable spread in ZFC?

A topological space has countable spread if every discrete subspace is at most countable.
By Theorem 8.10 in Todorcevic's book "Partition Problems in Topology", PFA implies that each regular space $X$...

**7**

votes

**1**answer

269 views

### Is a Borel image of a Polish space analytic?

A topological space $X$ is called analytic if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$.
We say that a topological ...

**11**

votes

**1**answer

389 views

### Is the dimension given by Klee trick ever sharp?

The Klee Trick allows one to find an $\mathbb{R}^m$ where two embeddings of same compact metric space have homeomorphic complements. More precisely, given two embeddings of a compact metric space $K$ ...

**7**

votes

**0**answers

231 views

### 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 ...

**8**

votes

**0**answers

204 views

### Topological applications of $\mathfrak{p}=\mathfrak{t}$

I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality.
Searching in ...

**6**

votes

**0**answers

115 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 $N\in\mathcal N$ ...

**1**

vote

**1**answer

71 views

### Is the map between mapping spaces, induced by the functor $\vert Sing(-)\vert$ continuous?

Let $X$ and $Y$ be topological spaces. Let $\vert Sing(-)\vert$ be the functor which sends a topological space to the (or "a"? there seem to be more possibilites, for me it's just important, that I ...

**11**

votes

**1**answer

499 views

### Frankl's conjecture restricted to finite topological spaces

A finite topological space is a finite family of finite sets that is closed under both union and intersection.
Frankl's conjecture states that for any finite union-closed family of finite sets, ...

**2**

votes

**0**answers

73 views

### A Baire space with meager projections

Question. Is there a Baire subspace $X$ of a Tychonoff power $M^\kappa$ of some separable metrizable space $M$ such that for any countable subset $A\subset \kappa$ the projection $$X_A=\{x{\...

**0**

votes

**1**answer

80 views

### On a pair of continuous functions “connected” by continuous functions

Suppose $X,Y$ are topological spaces with $Y$ homogeneous and $f,g:X\to Y$ continuous such that there exist continuous functions $u,v:Y\to Y$ such that $$f = u\circ g \text{ and } g= v \circ f.$$
...

**13**

votes

**1**answer

360 views

### Limit of homeomorphisms from square to square

Let $\square=[0,1]\times[0,1]$ be the unit square
and $f\colon\square\to \square$ is a continuous map that fixes the points on the boundary.
Assume $f$ is a limit of homeomorphisms $\square\to \...

**0**

votes

**1**answer

63 views

### Maximum of a sum of Gaussian functions

Consider the function which maps $\mathbb{R}^n$ to $\mathbb{R}$
\begin{align}
f(x) = \sum_{i=1}^{n} b_i\phi_i(x)
\end{align}
where $\phi_i(x) = \exp(-\frac{||x-x_i||_2^2}{2})$ are Gaussian functions ...

**0**

votes

**1**answer

64 views

### strict topology on multiplier algebras

Suppose $A$ is a $C^*$ algebra,$M(A)$ is the multiplier algebra.If $S$ is a subset of $M(A)$ which is compact for the strict topology on $M(A)$,is $S$ also a subset of $M(M(A))$ which is compact for ...

**6**

votes

**0**answers

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-...

**3**

votes

**1**answer

363 views

### If non-empty player has a winning strategy in Banach-Mazur game BM(X), then it also has in BM(Y)?

Let $f:X\rightarrow Y$ be a continuous, open, surjection function and second player (non-empty) has a winning strategy (not important which one, say for simplicity stationery st.) in $BM(X)$. Then can ...

**2**

votes

**0**answers

55 views

### A possible characterization of stratifiable spaces?

Let us recall that a regular topological space is semi-stratifiable if each point $x\in X$ has a countable family of neighborhoods $(U_n(x))_{n\in\omega}$ such that each closed subset $F\subset X$ is ...

**16**

votes

**1**answer

720 views

### Can one determine the dimension of a manifold given its 1-skeleton?

This may be an easy question, but I can't think of the answer at hand.
Suppose that I have a triangulated $n$-manifold $M$ (satisfying any set of conditions that you feel like). Suppose that I give ...

**2**

votes

**1**answer

73 views

### Every $b$-discrete space $X$ with countable injective weight is basically disconnected?

Recall that a space $X$ is called basically disconnected [1] if every cozero-set has an open closure.
According to Tkačuk [2], a space $X$ said to be $b$-discrete if every countable subset of $X$ is ...

**5**

votes

**1**answer

253 views

### Topology of connected subsets of the $3$-torus

Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$.
We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded.
I am ...

**4**

votes

**0**answers

64 views

### Is the Baireness a 3-space property of topological groups

It is known that the product of two Baire spaces can be meager.
On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably ...

**6**

votes

**0**answers

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 ...

**14**

votes

**1**answer

384 views

### A parametric version of the Borsuk Ulam theorem

Is there a topological space $X$, which is not a singleton, and satisfies the following property?
For every continuous function $f: X\times S^2\to\mathbb{R}^2$ there exist a point $x\in S^2$ such ...

**4**

votes

**0**answers

172 views

### homotopy type of box topology.

Suppose that $X$ is weakly equivalent to a point. Let $I$ be a set. Does $\prod_{i\in I}X$ weakly equivalent to a point, where $\prod_{i\in I}X$ is equipped with box topology ?

**10**

votes

**1**answer

476 views

### Homeomorphic characterization of the real line? [duplicate]

Let $A$ be a path-connected subset of $\mathbb R^2$ such that the removal of any singleton from $A$ splits $A$ into two open connected components, each of which is path-connected.
Is $A$ necessarily ...

**4**

votes

**1**answer

117 views

### What can say about $2^X= \{A\subseteq X: A\text{ is closed set} \}$, when $(X, \mathcal{U})$ is a compact uniform space?

It is known that if $(X, d)$ is a compact metric space, then hyperspace $2^X= \{A\subseteq X: A\text{ is closed set} \}$ is a compact space with Hausdorff metric
What can say about $2^X= \{A\...

**1**

vote

**3**answers

121 views

### Extension of continuous map on metric space

Let $X$ be a compact metric space, $A\subset X$ a closed subset and $f:A\to A$ be a continuous map.
Can $f$ be extended to a continuous map $X\to X$?
If so, is there an extension which is injective if ...

**2**

votes

**1**answer

109 views

### First countable geometric realization of a simplicial group

Suppose we have a simplicial group $G$.
What do we need from $G$ to get first countable $BG$, where $BG$ is a geometric realization of $G$?

**9**

votes

**3**answers

324 views

### Countable connected space where removing $1$ point destroys connectedness

Is there a countable connected space $(X,\tau)$ such that for all $x\in X$ the space $X\setminus\{x\}$ is not connected any more with the induced subspace topology?

**33**

votes

**1**answer

938 views

### Does there exist a continuous 2-to-1 function from the sphere to itself?

I am interested in the following question:
Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$?
I suspect the answer is no, but I don't know ...

**0**

votes

**1**answer

287 views

### Topological properties of complex valued Riemann sum limit curve and a particular integral inequality

I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...

**3**

votes

**2**answers

228 views

### Connected topological space $X$ such that $\emptyset, X$ are the only open connected subsets

Let $(X,\tau)$ be connected such that $\emptyset$ and $X$ are the only open connected subsets. Does this imply that $\tau = \{\emptyset, X\}$?

**5**

votes

**1**answer

117 views

### Uniqueness of limits and compactness implies closure

It is not difficult to prove that in a Hausdorff topological space every compact set is closed, and almost trivial that if in a topological space X every compact set is closed then X is T1. As ...

**2**

votes

**0**answers

51 views

### Refining monotone-light factorizations

Let $f:X\to Y$ be a continuous map between topological spaces. Consider the quotient map $\pi:X\twoheadrightarrow X/E$ given by decomposing the fibers of $f$ to their connected components.
In Lemma 6....

**4**

votes

**1**answer

117 views

### Is the category of inclusion prespectra bicomplete?

Working in compactly generated weak Hausdorff spaces, is the category of inclusion prespectra bicomplete?
I should probably specify that by inclusion prespectra, I mean prespectra such that the ...

**1**

vote

**0**answers

561 views

### A new generalization of the dimension?

During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...