# 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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### 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 ...

**4**

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**1**answer

126 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 ...

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87 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 ...

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**1**answer

65 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 $\mathcal{C}(\phi)$...

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**1**answer

438 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 ...

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166 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 ...

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85 views

### Topological groups containing the Sorgenfrey line

The Sorgenfrey line $\mathbb S$ is the real line endowed with the topology generated by the base consisting of all half-intervals $[a,b)$ for real numbers $a<b$.
The Sorgenfrey line is first-...

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76 views

### A special topological space [on hold]

Let $X$ be a compact topological space (not necessarily Hausdorff). I am looking for a charactrization the following property:
Property: Every closed subset $C$ of $X$ can be written as a disjoint ...

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**1**answer

65 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 ...

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69 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{\...

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665 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, \...

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**1**answer

73 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 $$f = u\circ g \text{ and } g= v \circ f.$$
...

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171 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 $X$...

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247 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 ...

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**1**answer

57 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 ...

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**1**answer

60 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 ...

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166 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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106 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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**1**answer

703 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 ...

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52 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 ...

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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 ...

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59 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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**1**answer

370 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 ...

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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 ?

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472 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 ...

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**3**answers

120 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 ...

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588 views

### What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...

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320 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?

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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\...

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**2**answers

222 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 $\tau = \{\emptyset, X\}$?

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919 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 ...

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61 views

### Invariant compact in division ring

Let $K$ be a discrete valued (with discrete valuation $v$) complete local division ring with ring of valuation $V$. Let $F$ be a compact subset of $V$. Suppose that for all $x\in F$ and for all $y\in ...

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138 views

### When Stone–Čech compactification is totally disconnected

A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets, and a topological space, $X$ is called completely regular exactly in case points can be ...

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39 views

### Difference between planar sub-continua and sub-continua on the surface $\mathbb{T}^2$?

Can anyone tell me what is the essential difference between planar sub-continua and sub-continua of the torus? I will appreciate if you can give me some references.

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49 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....

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**1**answer

143 views

### Sheaves on solenoids

Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $...

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**1**answer

86 views

### connected and quasi-connected separators of a space

Does there exist a connected topological space $X$ and a subset $A\subseteq X$ such that no connected component of $A$ separates $X$, but some quasi-component of $A$ separates $X$?
Meaning $X\...

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**1**answer

133 views

### Continuous binary operations on $\beta\mathbb{N}$

It is well-known that the operation of addition of two ultrafilters on the set $\mathbb{N}$ of natural numbers which extends the natural addition on $\mathbb{N}$ to $\beta\mathbb{N}$, the Cech-Stone ...

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134 views

### Is a plane set still metrizable if two new subsets are declared open?

I am thinking of forming a finer topology on a particular subset of the plane. Let $X\subseteq \mathbb R ^2$ be endowed with the Euclidean topology $\tau$. Let $A,B\subseteq X$. Let $\tau'$ be the ...

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256 views

### If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?

It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle?
...

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559 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 ...

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64 views

### Name for mappings that are “not quite projections”

Is there a known name for the following definition?
Consider topological spaces $X$, $Y$ and $f: X \rightarrow Y$ a continuous mapping. Then, $f$ is an "almost projection" if there is a topological ...

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**1**answer

110 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 ...

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97 views

### Slightly finer topology vs a quasi-component

Let $(X,\tau)$ be a topological space, and let $Q$ be a quasi-component of $X$. Let $S$ be a subset of $X\setminus Q$. Then is $Q$ necessarily a quasi-component of $X$ in the topology generated by $\...

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95 views

### Embeddability into $\beta\omega$ and $\omega^*$

It is well known that under CH every totally-disconnected compact F-space of weight at most $\omega_1$ can be embedded into the remainder $\omega^*=\beta\omega\setminus\omega$ of the Cech-Stone ...

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**1**answer

102 views

### Non-discrete $T_2$-space $(X,\tau)$ with $2^{|X|}$ retracts

If $(X,\tau)$ is a topological space, we call $A\subseteq X$ a retract if there is a continous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$ (we assume $A$ to be endowed with the subspace ...

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**1**answer

105 views

### Extending continuous functioms defined on the irrationals

Lavrentieff proved a Theorem which implies that every real valued continuous function defined on a dense subset $D\subseteq \mathbb R$ admits a continuous extension to some $G_\delta $ subset of $\...

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66 views

### $T_1$ version of Engelking theorem?

Theorem 6.1.23 in Engelking's Topology book says that in a compact space $X$ each quasi-component is connected. Quasi-component means the intersection of all closed-and-open subsets of $X$ containing ...

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187 views

### Rothberger property for finite covers

Let us recall that a topological space $X$ has the Rothberger property if for any sequence $(\mathcal U_n)_{n\in\omega}$ of open covers of $X$ there exists a sequence $(U_n)_{n\in\omega}\in\prod_{n\...

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89 views

### Slightly finer topology on a connected space

Let $(X,\tau)$ be a connected Hausdorff space.
Suppose $S\subseteq X$ is such that for every $U\in\tau$, $$U\cap S\neq\varnothing \implies U\cap \overline S\setminus S\neq\varnothing.$$
Is it ...