# 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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129 views

### Order convergence vs topological convergence in partially ordered sets

Short version of the question. If $(P,\leq)$ is a partially ordered set (poset), a topology denoted by $\tau_o(P)$ can be defined (see below). There is also another notion of convergence, called order-...

**4**

votes

**2**answers

88 views

### A reasonable topology on the group of minimal usco maps

An usco map is an abreviation for an upper semicontinuous multi-valued map with non-empty compact values. An usco map $f:X\multimap \mathbb R$ is called minimal is it coincides with each usco map $g:X\...

**14**

votes

**1**answer

642 views

### Homotopy pullback of a homotopy pushout is a homotopy pushout

Let's assume that we have a cube of spaces such that everything commutes up to homotopy.
The following holds:
- The right square is a homotopy pushout and
- all the squares in the middle are ...

**1**

vote

**1**answer

90 views

### Gluing locally defined continous functions over complex domain

This is a cross-post to the question I asked at MSE over almost a month ago.
Suppose $n, l, m \in \mathbb N$ and $n \ge l > m$. Let $T: \mathbb C \to \mathcal M(n \times l; \mathbb C)$ be ...

**1**

vote

**1**answer

187 views

### Question on K.Gobel's paper 1969

Let $X$ be uniformly convex Banach space. $f:K\rightarrow K$, such that $\parallel fx-fy\parallel \leq\parallel x-y\parallel\,\,\forall x,y\in K $, with $K$ a nonempty, closed, convex, bounded subset ...

**8**

votes

**1**answer

220 views

### Question about taking the Zariski closure in $\mathbb{A}_{\mathbb{R}}^n$

Let $\mathbb{A}_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with the Zariski topology, where closed sets are algebraic sets (in $\mathbb{R}^n$) defined by real polynomials.
Suppose $V \subseteq \mathbb{...

**3**

votes

**0**answers

53 views

### (When) is the Dedekind-MacNeille completion of a po-set Hausdorff?

Let $X$ be a p.o. Consider the topology on $X$ generated by
$$U_{x}^{-}:=X\setminus (x\uparrow),\quad U_{x}^{+}:=X\setminus (x\downarrow), \quad x\in X$$
Throughout this discussion I shall refer to ...

**1**

vote

**0**answers

71 views

### Is there any characterization for lifting clopen subsets

Let $Y$ be a subset of a topological space $X$. We say that a clopen subset $L$ of $Y$ lifts to $X$ whenever there exists a clopen subset $H$ of $X$ such that $H\cap Y=L$.
Let $X$ be a compact and $...

**1**

vote

**1**answer

79 views

### Borel $\sigma$-algebra on the space of Hölder continuous functions

Let
$(M,d)$ be a separable metric space
$E$ be a $\mathbb R$-Banach space
$\alpha\in(0,1]$
Moreover, let $$\left\|f\right\|_{C^{0+\alpha}(K,\:E)}:=\sup_{x\in K}\left\|f(x)\right\|_E+\sup_{\substack{...

**4**

votes

**1**answer

92 views

### Are homogeneous $T_2$-spaces flexible?

We say that a topological space $(X,\tau)$ is flexible, if for every closed discrete subset $D\subseteq X$ and every map $f: D\to X$ there is a continous map $f^X:X\to X$ such that $f^X|_D = f$.
$\...

**6**

votes

**1**answer

290 views

### iterated limit sets of a countable subset of real numbers

Let $A\subset \mathbb{R}$ be a closed subset, and $A'$ be the sets of limit points. We know that if $A$ is a countable set, $A'$ is a proper subset of $A$. Is it possible to find a subset( closed and ...

**6**

votes

**1**answer

220 views

### Is restriction a closed map?

Originally asked on MSE.
Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ ...

**2**

votes

**0**answers

67 views

### Lifting clopen subsets

Let $A$ be a subset of a topological space $T$, we say that clopen subset of $A$ lift to $T$ whenever $L$ is a clopen subset of $A$ the there exists a clopen subset $H$ of $T$ such that $H\cap A=L$.
...

**3**

votes

**0**answers

29 views

### Closedness of the partial order in complete Hausdorff semitopological semilattices

First some definitions.
A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the ...

**24**

votes

**1**answer

2k views

### Closed balls vs closure of open balls

We work in a separable metric space $(X,d)$. With $\overline{B}(x,r)$ I denote the closed ball around $x$ of radius $r$, and with $cl \ B(x,r)$ I denote the closure of the open ball. Clearly, we ...

**2**

votes

**0**answers

71 views

### Square Peg Problem and curve density

Square Peg Problem (or conjecture) is so famous. See this article
Let $CS:=\{\gamma:S^1\longmapsto\mathbb{R}^2 | \;\;\text {Square Peg Problem is true}\}$ and $C=\{\Upsilon:S^1\longmapsto\mathbb{R}^2 ...

**3**

votes

**1**answer

138 views

### Partitions of unity in constructive mathematics

Can someone point me to any substitutes for the partition of unity in Bishop's constructive mathematics?
In particular, under what circumstances can we construct a partition of unity subordinate to ...

**4**

votes

**3**answers

256 views

### Metrizable subspaces of separable spaces

Are metrizable subspaces of separable spaces separable?
Certainly subspaces of separable metrizable spaces are separable but subspaces of separable spaces need not be separable in general.

**1**

vote

**1**answer

85 views

### Compactness of the Fell topology and local compactness

Given some topological space $\mathbf{X}$, we consider the Fell topology on the set of closed subsets of $\mathbf{X}$. This is generated by sets of the form $I_U = \{A \mid A \cap U \neq \emptyset\}$ ...

**1**

vote

**0**answers

140 views

### Clopen subsets of a closed subspace of a spectral space

Let $X$ be a topological space. Set
$K(X) := \{ A\subseteq X\mid A$ is quasi-compact and open $\}.$ A topological space $X$ is called spectral,
if it satisfies all of the following conditions:
1) $...

**13**

votes

**2**answers

532 views

### Topological obstructions to existence of immersion

Let $M$ be a smooth, non-compact manifold.
a) Can one always find a smooth, compact manifold $N$ with $\dim(N) = \dim(M)$ and a smooth embedding $i: M \to N$ ?
b) If not, are there some concrete ...

**1**

vote

**1**answer

83 views

### Does this collection of properties imply metrizable?

I am working with a space with the following properties, and want to know if it is necessarily metrizable.
countable union of compact nowhere dense sets
T$_4$$=$T$_1$+normal
separable
Lindelöf
I ran ...

**1**

vote

**0**answers

55 views

### Comparing Different Notions of Unicoherence in the Plane

Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...

**8**

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**0**answers

191 views

### Is there a computable homeomorphism between two different Cartesian powers of the computable real numbers?

It's well know that it is surprisingly difficult to prove that $\mathbb{R}^n$ and $\mathbb{R}^m$ are not homeomorphic for $n\neq m$. Commonly proofs go through Brouwer's fixed point theorem, which is '...

**3**

votes

**1**answer

132 views

### Quotient of compact metrizable space in Hausdorff space

Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...

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vote

**0**answers

91 views

### Dual of $C(X)$ with the compact open topology

Let $X$ be a completely regular space and let $C_k(X)$ be the space of all continuous functions with the compact-open topology. If $X$ is completely metrizable, is the strong dual $C(X)^*$ the strong ...

**6**

votes

**1**answer

154 views

### Can one construct a regular neighborhood without an ambient space?

If I understand my PL topology correctly (and please correct me if I don't), if $K$ is a $k-$complex and $n\ge 2k+2$, then any two PL embeddings $a,b\colon K\to \mathbb{R}^n$ are isotopic. Therefore, ...

**2**

votes

**1**answer

227 views

### What are some surprising facts that happen after you remove a point to a space? [closed]

There are some facts that are really impressive after you remove a point to a space. Some typical examples are the existence of exotic spheres or the fact that
$S^4$ is not almost complex. Or some not ...

**3**

votes

**1**answer

111 views

### Nice representation of open sets in $\sigma$-algebras in certain circumstances

Let $(X,\tau)$ be a topological space. For a given topological base $\mathcal{E}$ for $\tau$, let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$.
Q. Assume ...

**4**

votes

**1**answer

84 views

### Nice arrangement of open sets in $\sigma$-algebras

Let $X$ be a topological space and $\mathcal{E}$ be a topological base for $X$. Let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$.
Q. Let $O$ be an open ...

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vote

**0**answers

65 views

### Does There Exist a Planar, Linear, Triodic Tree-Like Continuum?

Motivated by Continuum image of line is chainable?
A planar continuum $X$ is a compact, connected subset of the plane. It is linear if there is a continuous bijection from $[0,1)$ onto $X$, for ...

**16**

votes

**1**answer

349 views

### Lowest Dimension for Counterexample in Topological Manifold Factorization

Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is ...

**3**

votes

**1**answer

116 views

### Has the Erdős space the structure of a monothetic topological group?

This question is motivated by this MO-problem asking if the Erdős spaces $\mathfrak E$ and $\mathfrak E_c$ admit a self-homeomorphism with dense orbits of points.
The affirmative answer would follow ...

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vote

**1**answer

96 views

### Continuum image of line is chainable?

Let $X$ be a metric continuum (compact + connected) which is the one-to-one continuous image of the interval $[0,\infty)$. Such an $X$ is called a linear continuum.
It seems like $X$ should be ...

**4**

votes

**1**answer

84 views

### Hereditary Lindelöfness in $C_p$-spaces

Let $X$ be a (infinite) separable topological space and consider $C_p(X)$, the space of continuous functions on $X$ endowed with the point-wise convergence topology.
Q. I am looking for ...

**3**

votes

**1**answer

84 views

### A question on quasitopological group

Suppose that $G$ is a regular feebly compact Moore quasitopological group. Must $G$ be a topological group? This was previously posted here on MathSE also.
A semitopological group $G$ is a group $G$ ...

**4**

votes

**1**answer

206 views

### Automorphisms of Erdös spaces

It is well-known that there is an automorphism of the Cantor set $h:C\to C$ such that $\overline{\{h^n(c):n\in \mathbb Z\}}=C$ for every $c\in X$.
In other words, there is a self-homeomorphism of $C$...

**3**

votes

**0**answers

63 views

### Compact subspace of sober space

We know from lemma 1.2.5 in part C of Sketches of an Elephant (by Johnstone) that both open and closed subspaces of a sober space are again sober. This raises the following question.
Question: Is a ...

**4**

votes

**1**answer

104 views

### Fréchet–Urysohn subspaces in $[0,1]^{[0,1]}$

A topological space $X$ is called a Fréchet–Urysohn space if for each subset $A\subseteq X$ and for each point in its closure, $x\in\overline{A}$, there is a sequence (not just a net, but a sequence) $...

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**0**answers

40 views

### A weaker locally convex topology on a pospace

A pospace is a topological space $X$ endowed with a closed partial order $\le$. A pospace $X$ is locally convex if it has a base of the topology consisting of open order-convex sets. A subset $A$ of a ...

**1**

vote

**1**answer

69 views

### Product of point-removal insensitive spaces

We say that a topological space $(X,\tau)$ is point-removal insensitive if for all $x\in X$ we have $X\cong X\setminus \{x\}$.
If $X,Y$ are point-removal insensitive, does this imply that $X\times Y$ ...

**2**

votes

**1**answer

326 views

### How “compact” are sets of finite measure?

Let $K$ be a compact set of $\mathbb R^n$, then every open cover of $K$ will have a finite subcover.
Now consider the following situation:
Everything I say in the following is with respect to the ...

**1**

vote

**0**answers

59 views

### An example of a Borel map of the first class

Let $X,Y$ be compact metric spaces, $2^X$ the set of all closed subsets of $X$ and $f:X\to Y$ be a 1st class Borel mapping.
Im trying to check Borel class of mapping $G:2^Y\to 2^X$. I submit it in a ...

**6**

votes

**1**answer

287 views

### Is $\mathbb{R}^\omega \cong (\mathbb{R}^\omega \setminus \{x\})$?

Is $\mathbb{R}^\omega \cong (\mathbb{R}^\omega \setminus \{x\})$, where $\mathbb{R}^\omega$ is given the product topology, and $x\in\mathbb{R}^\omega $?

**5**

votes

**1**answer

292 views

### Is the set $\{\zeta: \rho(A(\zeta))< 1\}$ connected for matrices under parameterization of first $m$ rows?

$\newcommand{\eqqcolon}{=\mathrel{\vcenter{:}}}$
Fix $n, m \in \mathbb N$ with $n > m$. Let $\zeta \in \mathcal{M}(m \times n; \mathbb C)$ and we fix a $\zeta_0 \in \mathcal M( (n-m) \times n; \...

**2**

votes

**1**answer

154 views

### Relation between the weak star topology and hereditary Lindelöfness

Let $X$ be a Banach space. Is the following implication valid?
$$ (X,w) \textrm{ is hereditarily Lindelöf}~ \Rightarrow X^*~ \textrm{is separable} $$
The converse is clearly true, since the ...

**4**

votes

**0**answers

91 views

### Is the closure $\overline{ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) < 1\} }$ equal to $ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) \le 1\}$

I am not sure whether this question fits this forum (I will delete it if not appropriate here). But I asked this on MSE over a week ago with no answer and then put a bounty still got no answer. Here ...

**2**

votes

**1**answer

311 views

### Punching a hole into $\mathbb{R}^\omega$

Let $\mathbb{R}^\omega$ be endowed with the product topology. Is there a nonempty open set $U\subseteq \mathbb{R}^\omega$ such that $\mathbb{R}^\omega\cong \mathbb{R}^\omega\setminus \text{cl}(U)$?
(...

**7**

votes

**0**answers

230 views

### Does geometric realization commute with passing to the compactly generated topology?

My question is in the title, but here is a more detailed formulation:
Let Top be the category of all topological spaces and continuous maps, and let CGTop be the subcategory of compactly generated ...

**3**

votes

**1**answer

201 views

### Example of a Baire Class $1$ function $f$ satisfying $\omega\cdot n<\beta(f)\leq \omega\cdot (n+1)$ for some natural number $n\geq 1.$

Definitions: Let $X$ be a Polish space (separable completely metrizable topological space).
A function $f:X\to\mathbb{R}$ is Baire Class $1$ if it is a pointwlise limit of a sequence of continuous ...