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19
votes
3answers
573 views

Connectedness in the language of path-connectedness

Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds? A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a ...
3
votes
1answer
148 views

Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space?

Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space? (You find the definition of $\mathcal{P}(\omega)/fin$ here.) Remark: According to this, the interval topology of ...
5
votes
1answer
114 views

Infinite Hausdorff space that is not homeomorphic to any proper quotient

Let $S$ be a set and $\vartheta$ be an equivalence relation on $S$. We say that $\vartheta$ is proper if there are $x\neq y\in S$ with $(x,y)\in\vartheta$. Is there an infinite Hausdorff space ...
8
votes
2answers
210 views

Images of $\{0,1\}^\kappa$

Is there a compact topological space $(X,\tau)$ such that for no cardinal $\kappa$ there is a surjective continuous map $e:\{0,1\}^\kappa \to X$? (We assume that $\{0,1\}$ is endowed with the ...
3
votes
2answers
193 views

Critical topological spaces

This is a follow-up question to Existence of injective neighborhood selection function as separation axiom. Let $(X, \tau)$ be a topological space. If there is an injective map $f:X\to\tau$ such that ...
4
votes
3answers
123 views

Existence of injective neighborhood selection function as separation axiom

Let $(X, \tau)$ be a topological space. We say that $(X,\tau)$ is $T_{\text{inj}}$ if there is an injective map $f:X\to\tau$ such that $x\in f(x)$ for all $x\in X$. It is not hard to see that $T_1$ ...
6
votes
3answers
229 views

Borel cross section

It is known from metric space topology that a closed equivalence relation on a Polish space has either countably many or $\mathfrak{c}$ many equivalence classes. A short elementary proof is given in ...
21
votes
3answers
701 views

Possible categorical reformulation for the usual definition of compactness

Let $X$ be a compact topological space, $f_i:Y_i\to X$ a family of continuous maps such that the topology on $X$ is final for it (i.e., $U\subset X$ is open iff $f_i^{-1}(U)$ is open for each $i$, for ...
6
votes
2answers
224 views

Product of “prime” topological spaces

We call a topological space $(X,\tau)$ product-decomposable if there is an index set $I$ and subsets $X_i\subseteq X$ for $i\in I$ such that $|X_i| > 1$ and $X \cong \prod_{i\in I} X_i$ where each ...
3
votes
1answer
197 views

Equivalence relation defined by the existence of a homeomorphism

Let $(X,\tau)$ be a topological space. We assign to $(X,\tau)$ an equivalence relation $\simeq_{(X,\tau)}$ in the following way: $x\simeq_{(X,\tau)} y$ if and only if there is a homeomorphism ...
2
votes
4answers
359 views

Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$

Let $(X,\tau)$ be a topological space. Let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$. What can be said about spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$? For ...
2
votes
1answer
106 views

Is $ {C_{c}}(G) $ a meager subset of $ {L^{2}}(G) $ for a second-countable locally compact Hausdorff group $ G $?

The following problem is a stumbling block in a research project that I am working on: Problem. Let $ G $ be a second-countable locally compact Hausdorff group with a fixed Haar measure. Is it ...
4
votes
2answers
517 views

Co-Hausdorffification

Given a topological space $(X,\tau)$ we can define the "$T_2$-ification" of $X$ by setting $T_2(X,\tau) = X/\simeq$ where $x\simeq y$ in $X$ if and only if for every open neighborhood of $x$ has ...
6
votes
0answers
98 views

Is $(\omega+1)^\omega$ with the box topology ultraparacompact?

Let $\omega+1$ be endowed with the interval topology, that is $U\subseteq (\omega+1)$ is open if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite. We call $U\subseteq (\omega+1)$ basic if ...
3
votes
1answer
152 views

Paracompact zero-dimensional space without clopen partition refinement

If $(X,\tau)$ is a topological space we say that an open cover $\mathcal{U}$ is a clopen partition cover if it consists of disjoint clopen sets. Trivially, every clopen partition cover is locally ...
6
votes
1answer
323 views

Sierpinski-like spaces

Let $\mathbb{S}$ be the Sierpinski space, that is $\mathbb{S}$ has $\{0,1\}$ as a base set, and $\tau = \{\emptyset, \{0\}, \{0,1\}\}$ as a topology. The Sierpinski space $\mathbb{S}$ has the ...
3
votes
0answers
130 views

The Hausdorff quotient of totally disconnected space

Let $G$ be a second countable locally compact Hausdorff group and $X$ be a second countable locally compact almost-Hausdorff $G$-space. If $X$ is totally disconnected and the orbit space X/G is ...
5
votes
2answers
182 views

“All retracts are closed” as separation axiom

The starting point of this question is the fact that any retract of a $T_2$-space is closed. Let's say a topological space $(X,\tau)$ is $T_{\textrm{rc}}$ if all retracts of $X$ are closed. All ...
4
votes
3answers
324 views

Infinite topological spaces such that every subset is a retract

Let $X$ be an infinite set and let $(X,\tau)$ be a topological space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does this ...
2
votes
1answer
128 views

Hausdorff spaces such that every subset is a retract

Let $(X,\tau)$ be a Hausdorff space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does $\tau$ have to be discrete?
3
votes
2answers
250 views

Is the defining bijection for a pullback of topological spaces a homeomorphism?

I work in the category of CGWH spaces enriched over themselves. If a space $P$ is the pullback of $A \rightarrow B \leftarrow C$, then for every space $T$ the canonical map $$Top(T,P) \rightarrow Top ...
0
votes
1answer
123 views

totally disconnected sets and homeomorphisms [closed]

For every totally disconnected perfect subset S in the plane one finds a homeomorphism of the plane onto itself mapping S onto the ternary Cantor set. This is an exercise in a book by Engelking and ...
6
votes
0answers
121 views

A forked plane continuum

I came up with this question while trying to solve the following MO one: Does every connected set that is not a line segment cross some dyadic square? Suppose $C$ is a plane continuum (i.e. a ...
7
votes
1answer
94 views

Universality with respect to quotients

Is there an infinite cardinal $\kappa$ for which the following statement (S) true? (S) : There is a topology $\tau_\kappa$ on $\kappa$ such that for all topological spaces $(X,\tau)$ with $|X|\leq ...
5
votes
2answers
95 views

scott continuity, sub additivity

Let $(X, \sqsubseteq_x)$ and $(Y, \sqsubseteq_y)$ be two posets and let $\delta_x:X \to X$ and $\delta_y:Y \to Y$ be two closure operators (monotone, inflationary, idempotent). Then, a monotone ...
1
vote
1answer
85 views

Can we build a continuous function from “fibers”/preimages defined over a topological base?

I am looking for some proof insight or literature references for a statement which, if it's actually true, is probably a pretty trivial thing. I hope the question has not been asked here before, and ...
14
votes
0answers
156 views

Finite union of closed convex sets is triangulable?

I posted this question on Stackoverflow, but didn't get an answer. Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that is, ...
3
votes
0answers
326 views

Topological proof of a result in Logic

I proved the result below using logic. My questions: Can this theorem be proved by purely topological means? Do you know any theorems that either can be used to prove the same result, or which give ...
10
votes
1answer
353 views

continuous images of open intervals

The well-known Hahn-Mazurkiewicz theorem characterizes those nonempty Hausdorff spaces $X$ that admit a continuous surjection $\alpha: [0, 1] \to X$ from the closed unit interval: it is necessary and ...
0
votes
1answer
85 views

Is a weakly separable group always Lindelöf?

By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...
2
votes
0answers
62 views

A construction with Hyperspace of continums

Let $X$ be a compact connected metric space. Its hyperspace is denoted by $2^{X}.$ $X$ is considered as a subset of $2^{X}$ via the embedding $x\mapsto \{x\}$. Assume that $f:X\to X$ is a ...
2
votes
1answer
289 views

A question about small cardinals related to Michael's Problem

I'm studying some applications of small cardinals related to the Michael's Problem. Recall that we say that a space $X$ is a Michael space if X is a regular Lindelöf space such that $X\times ...
0
votes
0answers
65 views

hyperspaces and selection principals

Two things bother me for which I haven't found an answer yet: 1.Is anyone familiar with an example of a topological space $X$, in which the hyperspace $2^X$ with the upper Fell topology is ...
1
vote
1answer
102 views

subspace in pseudotopological space

Every topological space gives rise to a pseudotopological space. Conversely, if $X$ is a pseudotopological space then we can define a topology on $X$ such that every filter converging to $x$ in the ...
5
votes
1answer
435 views

sets without perfect subset in a non-separable completely metrizable space

Suppose $X$ is a completely metrizable (but not separable) space. Suppose $D$ is a Borel (actually $F_{\sigma}$) subset of $X$. Is there any logical relation between the following statements? [1] ...
4
votes
0answers
143 views

Is the limit set of a group action always closed?

Let $G$ be a discrete group acting on a compact metric space $X$. A point $x\in X$ is called a limit point, if there is a base point $x_0\in X$ and an injective sequence $(x_k)_{k\in\mathbb{N}}$ in ...
5
votes
1answer
220 views

Nonmetrizable compact totally disconnected spaces without isolated points

Brouwer proved that a topological space is homeomorphic to the Cantor set if and only if 1) it's non-empty, 2) it's compact, 3) it's totally disconnected, 4) it has no isolated points, and 5) ...
3
votes
1answer
216 views

Continuous Functions

Is there a pair of continuous surjective functions say $f_1$ and $f_2$ from $\mathbb{Q}$ to itself such that for every $x, y \in \mathbb{Q}$, $f_1^{-1}(x) \cap f_2^{-1}(y)$ is non-empty?
1
vote
3answers
201 views

Axiomatization of locally compact Hausdorff spaces via compact subspaces

The usual axiomatization of a topological space (in the sense of Bourbaki) goes by declaring certain subsets as being open and such that a few axioms are fulfilled by the family of open subsets. It ...
4
votes
2answers
232 views

An approximate infinite-dimensional fixed point theorem

Given $\epsilon > 0$ and $f : [0, 1]^{\omega} \rightarrow [0, 1]^{\omega}$, can we find $x$ such that $x \in \textrm{Conv}\left( \left\{f(y) : ||y - x||_{\infty} < \epsilon\right\}\right)$? In ...
3
votes
2answers
198 views

A set intersecting the graph of any continuous function in a finite set

New version of the problem I am looking for a characterization of those completely regular and hausdorff spaces $X$ such that the follwing is true: If $f :X\longrightarrow \Bbb{R}$ is continuous ...
5
votes
0answers
278 views

Sequences and pseudocharacter in compact spaces

Is there a consistent example of a compact Hausdorff space $X$ on which the following holds? i) there is a $Y \in {[X]}^{\aleph_1}$ such that $\psi (Y) = \aleph_1$; and ii) there is no non-trivial ...
5
votes
2answers
474 views

A question about the Stone–Čech compactification of discrete spaces

Let $D(\kappa)$ be the discrete space of cardinality $\kappa$, and $\beta D(\kappa)$ its Stone–Čech compactification. Is there, for every infinite cardinal $\kappa$, a subset $Y \in [\beta ...
9
votes
0answers
282 views

Linearly Lindelöf spaces with Lindelöf degree of uncountable cofinality

Is there a linearly Lindelöf space $X$ with $\operatorname{cf} (L(X))> \aleph_{0}$ (where $L$ is the cardinal function Lindelöf degree)? $L(X)$ must be a limit cardinal, like $\aleph_{\omega_{1}}$ ...
8
votes
1answer
350 views

Is there a $P$-space linearly Lindelöf and non-Lindelöf?

A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersection). A topological space $X$ is linearly ...
1
vote
1answer
135 views

A space with countable tightness which is not a Fréchet space?

I need a space with countable tightness which is not a Fréchet space. In this space, I am searching for a point with no deleted neighborhood consisting entirely of P-points. (A P-point is a point $x ...
17
votes
0answers
489 views

Can closed compacts in a topological group behave “paradoxically” with respect to unions, intersections, and one-sided translations?

Consider two closed compacts $A$ and $B$ in a topological group $\Gamma$. Let $A'$ be a left translation of $A$ and $B'$ a left translation of $B$: $A' = aA$, $B' = bB$. Suppose it is known that ...
11
votes
1answer
841 views

Analysis of the boundary of the Mandelbrot set

Motivation: The Mandlebrot set is a simply connected set with an infinitely complex boundary, but CAN one move from interior to the exterior of this topological space by just crossing over a finite ...
0
votes
1answer
368 views

Property of Mrowka

A topological space $X$ satisfies Property $K_1$ (Property of Mrowka) if the closure of the union of arbitrarily many $G_\delta$ sets of $X$ coincides with its sequential closure (the sequential ...
6
votes
1answer
289 views

On the cardinality of perfect spaces with the countable chain condition

QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spaces? Recall that a ...