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19
votes
3answers
574 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 ...
5
votes
0answers
425 views

Continuous images of Cantor cubes

The original title of this question was "Is there only one (up to homeomorphism) zero-dimensional homogeneous dyadic space of weight $\mathfrak{c}$?". I changed it with the hope of getting a bit more ...
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 ...
6
votes
3answers
230 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 ...
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 $T_1$ ...
21
votes
3answers
702 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 ...
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 ...
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
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 ...
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 ...
50
votes
4answers
3k views

How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. ...
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 ...
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 ...
15
votes
5answers
1k views

Compactness of the Hilbert cube without the Axiom of Choice

I am just curious: is there a published proof of the compactness of the Hilbert cube that does not use the Axiom of Choice, or is it well known?
6
votes
6answers
3k views

Is a topology determined by its convergent sequences?

Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the ...
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?
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 ...
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 ...
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 ...
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
5answers
3k views

Locally compact Hausdorff space that is not normal

What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of ...
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 ...
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 ...
14
votes
6answers
2k views

Topological characterization of the closed interval $[0,1]$

This question is related to question 92206 "What properties make $[0, 1]$ a good candidate for defining fundamental groups?" but is not exactly equivalent in my opinion. It is even suggested in one ...
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 ...
1
vote
3answers
828 views

Zero-dimensional space

Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A_{\alpha} : \alpha\in S\}$ such that ...
5
votes
1answer
436 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] ...
1
vote
1answer
462 views

Does there exist a countable partition of [0,1] by disjoint closed subsets? [duplicate]

Possible Duplicate: Why are the integers with the cofinite topology not path-connected? As in the title, is it possible to find closed, disjoint subsets $C_n$ of $[0,1]$ such that $[0,1] = ...
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 $\aleph_{\omega_{1}}$ ...
0
votes
4answers
3k views

Does Cauchy continuity imply uniform continuity? [No.] [closed]

It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff $$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...
1
vote
1answer
517 views

Example of a topological space

In my recent research, I defined a topological space $X$ to be an $EZ$-space if for every open subset $A$ of $X$, there exists a collection $\{A_{\alpha}: \alpha\in S\}$ of clopen subsets of $X$ such ...
-1
votes
1answer
366 views

Fuzzy topology : references [closed]

Hey. I'm looking for references in fuzzy topology. Does anyone know a good book ?
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
1answer
217 views

Relative extremely disconnected space

A topological space $X$ is called relative extremely disconnected if it has a base $B$ (for open subsets) such that disjoint elements in $B$ have disjoint closure. Does it exist an infinite ...
0
votes
1answer
379 views

Extremely disconnected space

A topological space $X$ is called relative extremely disconnected if it has a base $B$(for open subsets) such that disjoint elements in B have disjoint closure, i.e, if $C, D$ in $B$ and $C\cap ...
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 ...
3
votes
2answers
1k views

Is $C^{\infty}[0,1]$ or $S$ separable?

I want to know if $C^{\infty}[0,1]$ or $S$ (Schwartz function space) is separable. Can somebody offer me some results or references? Thank you!