# Questions tagged [gn.general-topology]

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

**11**

votes

**1**answer

335 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

**2**answers

129 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

**1**answer

44 views

### Two notions of boundedness in metrizable topological vector space

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

**1**answer

75 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

**0**answers

37 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

**0**answers

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 ...

**-2**

votes

**0**answers

237 views

### Why is the notion of compactness so powerful? [closed]

According to you, what are the deep reasons, at a very fundamental level, that makes the notion of compactness so useful and ubiquitous throughout modern mathematics: theory of compact groups, compact ...

**5**

votes

**1**answer

93 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 $|G|$ ...

**0**

votes

**0**answers

275 views

### Analogs of “Intersection Forms” for 3-manifolds and 5-manifolds [closed]

We recall the intersection form for 4-manifolds $M^4$,
$$Q_{M^4}: H^2(M^4;\mathbb Z)\times H^2(M^4;\mathbb Z) \to \mathbb Z$$
defined by
$$
Q_{M^4}(a,b) =\langle a\cup b,[M^4]\rangle=\int_{M^4} ab.
$...

**4**

votes

**2**answers

203 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

**1**answer

51 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

**1**answer

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

**1**answer

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

**1**answer

135 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**

vote

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

105 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)?

**6**

votes

**0**answers

83 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

**1**answer

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

**0**answers

182 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

**1**answer

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

**2**answers

653 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

**1**answer

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

**0**answers

186 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

**1**answer

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

**1**answer

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**

votes

**0**answers

58 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

**2**answers

282 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

**1**answer

112 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

**2**answers

182 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

**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

92 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

**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 ...

**4**

votes

**1**answer

261 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

**0**answers

120 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

**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 ...

**5**

votes

**1**answer

151 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

**0**answers

109 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

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

**9**

votes

**1**answer

510 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

**0**answers

203 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

**1**answer

304 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

**1**answer

70 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

**0**answers

72 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

**1**answer

689 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

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

**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

266 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 ...

**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 ...