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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.

2
votes
0answers
37 views

Norm closure of $C_b^1(\mathbb{R})$

I want to determine what the closure of $C_b^1(\mathbb{R})$, the space of continuous differentiable functions with bounded derivative, with respect to the supremums norm is. I think that $\overline{...
3
votes
0answers
52 views

Contractible $T_2$-space without subspace homeomorphic to it

This is an update to an older question. Is there a contractible $T_2$-space $(X,\tau)$ on more than $1$ point such that no proper subspace of $X$ is homeomorphic to $X$?
-4
votes
2answers
160 views

Does every connected $T_2$-space $X$ have a proper subspace homeomorphic to $X$? [on hold]

Does every connected $T_2$-space $(X,\tau)$ with $|X|>1$ have a proper subspace homeomorphic to $X$?
6
votes
0answers
108 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? ...
12
votes
1answer
362 views

Are Hausdorff measures on the real line Haar measures for some locally compact topology?

For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...
3
votes
2answers
133 views

Rigid space, but with homeomorphic neighborhoods

What is an example of a topological space $(X,\tau)$ on more than one point, with the following properties? the only homeomorphism from $X$ to itself is the identity, and given $x,y\in X$ there are ...
-4
votes
1answer
45 views

Two notions of boundedness in metrizable topological vector space [on hold]

In a metrizable topological vector space X with the metric d, a subset A is said to be bounded if it can be absorbed by any neighbourhood of 0 and a subset A is said to be d-bounded if its diameter ...
1
vote
1answer
78 views

Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive

Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty. Let $S : X → X$ and ...
2
votes
0answers
38 views

Metrically homogeneous spaces as inverse limits

Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following: Is $X$ ...
1
vote
0answers
68 views

Minimal size of an open affine cover for an open complement

Let $X$ be a smooth projective scheme and $Y$ be a projective subscheme of $X$, not necessarily smooth. Are there any known results about the minimal size of an open affine cover (number of affines in ...
5
votes
1answer
95 views

Does every bijective graph endomorphism restrict to a full-cardinality isomorphism?

Given a graph $G$, and a bijective endomorphism $f$ (that is, a graph homeomorphism $f : G \to G$ that establishes a bijection on the vertices), it is true that $f$ is an automorphism whenever $|G|$ ...
4
votes
2answers
204 views

Set of topologies on a group making it a compact Hausdorff topological group

Maybe stupid, but from the following well known facts about compact Hausdorff (CH) spaces: CH topologies on a given set are pairwise incomparible (one is not finer or coarser than the other). There ...
0
votes
1answer
53 views

Basis or subbasis for Scott topology

Let $X$ be a partially ordered set. A subset $S\subseteq X$ is called Scott-open if and only if it is: Upward-closed: $x\in S$ and $x\le y$ implies $y\in S$; Inaccessible by directed suprema: if $D\...
0
votes
1answer
49 views

Empty interior lack of minima

Suppose that $U \subseteq \mathbb{R}^d$, and satsifies $U$ is dense in $\mathbb{R}^d$, U has empty interior, Then is it possible that $$ \inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x), $$ ...
2
votes
1answer
45 views

Is the Scott topology generated by the ideals as the closed sets?

Let $X$ be a directed-complete partial order, or even a complete lattice. A subset $S\subseteq X$ is called Scott-closed if and only if it is: Downward-closed: $y\in S$ and $x\le y$ implies $x\in S$; ...
6
votes
1answer
136 views

Geometry of complements to compacts of codimension 2

Let $K\subset \mathbb{R}^n$ be a (nonempty) compact of covering dimension $\le n-2$. In particular, $K$ does not separate $\mathbb{R}^n$ (even locally). I will equip $M=\mathbb{R}^n-K$ with the ...
1
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0answers
28 views

Localized connected expansions

Given a connected space, it is easy to tell if there is a connected expansion because maximal connected spaces (those admitting no finer connected topology) have the property that every dense subset ...
3
votes
0answers
43 views

Galois Covering induces new Cover $Ind_H ^G(Y)$

I have a question about the construction of the so called "induced cover" introduced in Tamas Szamuely's "Galois Groups and Fundamental Groups" (see page 84): We consider a group $G$ which contains a ...
1
vote
1answer
72 views

Non-homeomorphic computable metric spaces whose computable points are computably homeomorphic

This is a follow-up of sorts to an earlier question on mine in that it should be easier to construct a positive example of this, if it exists. To be clear about definitions, a computable metric space ...
1
vote
1answer
106 views

Separability of the Stone space of a free sigma-algebra

Let $X$ be the Stone space of the free $\sigma$-algebra $A$ on $\omega_1$ free generators. Is $X$ separable (i.e. does $X$ contain a countable dense set)?
7
votes
0answers
86 views

Measure support decomposition that “tends to infinity”

I would like to know the answers to the following two questions. Let $S$ be a locally compact Hausdorff space, $\mu$ be a regular Borel measure with non-compact support $M$. Denote $$ \mathscr{H}=\{\...
6
votes
1answer
234 views

Covering compact Hausdorff spaces with closed $G_\delta$ sets

I'm thinking about results of the form: Under assumption $A$, if $X$ is a compact Hausdorff space and $C$ is a cover of $X$ by closed $G_\delta$ sets, then there is a subcover of cardinality $\leq\...
9
votes
0answers
183 views

Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$

An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane. The following questions are motivated by Anton Petrunin's Disc bounded ...
2
votes
1answer
197 views

Fixed point property and interval topology

Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\leq ...
18
votes
2answers
661 views

Simple connectedness under a metric undistortion condition: on a tricky point in an argument of Gromov

The context I have been reading Gromov's Metric Structures..., and came upon result 1.14.(a), page 11, which states the following. Let $K\subset\mathbb R^d$ be a compact subset, and $d_\ell$ its ...
6
votes
1answer
125 views

Name for topological spaces where “every point has a local base wellordered by reverse inclusion”?

There are many properties regarding local bases of a topological space, like first countable if every point has a countable local base. Is there a similar name for a space where "every point has a ...
12
votes
0answers
190 views

Counter example to lifting contractibility of a topological space

I'm looking for a simple example of an open proper continuous map between topological spaces $\varphi:X\to Y$ such that : $Y$ is contractible and locally contractible ; for any $y\in Y$, $\varphi^{-1}...
5
votes
1answer
132 views

Chromatic number of a connected Hausdorff space

Let $(X,\tau)$ be a topological space such that $\tau$ contains no singleton. We say that a map $c:X\to \kappa$, where $\kappa$ is a cardinal, is a coloring for $(X,\tau)$, if for every $U\in \tau\...
1
vote
1answer
55 views

Does exponential law bijective implies evaluation map continuous?

Husemoller in his "Fibre Bundles" writes that the exponential law $$ \theta \colon B^{A \times X} \to (B^X)^A $$ which is always injective, is bijective if and only if the evaluation map $$ Ev \colon ...
2
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0answers
59 views

Is there a normal separable sequential $\aleph$-space with uncountable extent?

It is a classical fact from the undergraduate course of General Topology that under CH (more precisely, under $2^{\omega_1}>\mathfrak c$) every separable normal space has countable extent, i.e., ...
8
votes
2answers
283 views

Are almost sequential spaces sequential?

A topological space $X$ is called $\bullet$ sequential if for each non-closed subset $A\subset X$ there exists a sequence $\{a_n\}_{n\in\omega}\subset A$ that converges to a point $a\notin A$; $\...
0
votes
1answer
113 views

Countable intersections in topological space

If a T1 topological space is closed under countable intersections, does this necessarily make the topology discrete? It is easy to construct a counterexample if the topological space is not assumed to ...
2
votes
2answers
185 views

Is there a second countable topological space, which can not be equipped with a finite borel measure of full support?

If I have a second countable topological space X, can i Always find a finite borel measure, such that every non-empty open set has positive measure? without second countability, the discrete topology ...
2
votes
1answer
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
1answer
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
1answer
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 ...
4
votes
1answer
263 views

Is each cosmic space cometrizable?

A regular topological space $X$ is called $\bullet$ cosmic if $X$ is a continuous image of a separable metrizable space; $\bullet$ cometrizable if $X$ admits a weaker metrizable topology such that ...
2
votes
0answers
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; ...
2
votes
0answers
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 ...
5
votes
1answer
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
0answers
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 ...
3
votes
1answer
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 $\mathcal{C}(\phi)$...
9
votes
1answer
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 ...
8
votes
0answers
205 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 ...
11
votes
1answer
305 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-...
1
vote
1answer
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 ...
2
votes
0answers
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{\...
2
votes
1answer
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, \...
0
votes
1answer
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 $$f = u\circ g \text{ and } g= v \circ f.$$ ...
9
votes
1answer
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 $X$...