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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.

4
votes
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
3answers
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
votes
0answers
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
1answer
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 ...
1
vote
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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 ...
1
vote
0answers
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
1answer
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
1answer
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 ...
1
vote
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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) $...
0
votes
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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 ...