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3 votes
1 answer
132 views

Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?

If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
Dominic van der Zypen's user avatar
3 votes
1 answer
165 views

Menger and Scheepers subsets of $\mathbb R$

$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...
Nur Alam's user avatar
  • 505
24 votes
2 answers
2k views

Why are extremally disconnected spaces so hard to give examples of?

Recall that an extremally disconnected space is a Hausdorff topological space in which the closure of any open set is still open. On the surface, this doesn't seem like a very remarkable condition ...
Gro-Tsen's user avatar
  • 32.5k
7 votes
0 answers
349 views

An open set which is not the union of a closed set and a countable set

The following fact is probably a known result: Fact. Let $X$ be an uncountable Polish space. Then there exists an open subset of $X$ which is not the union of a closed set and a countable set. Proof:...
Paolo Leonetti's user avatar
5 votes
2 answers
247 views

Definability properties of box-open subsets of Polish space

Let $X$ be a perfect Polish space $X$, so that $X^\omega$ is also a Polish space under the product topology. Call a subset $\mathcal{X} \subseteq X^\omega$ box-open if it is an open subset of $X^\...
Clement Yung's user avatar
  • 1,412
5 votes
1 answer
183 views

What is the extent of a $\Sigma$-product of a (uncountable) power of a (countable) discrete space?

Recall that a $\Sigma$-product of a family of spaces $\{X_s:s\in S\}$ with a base point $a=(a_s)\in \prod_{s\in S} X_s$ is the subspace $$\Sigma(a)=\{x\in \prod_{s\in S} X_s: |\{s\in S:a_s\neq x_s\}|\...
J. Casas's user avatar
  • 308
6 votes
1 answer
149 views

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
Alexander Osipov's user avatar
3 votes
1 answer
177 views

Is there a metric separable space with the following properties...?

Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$. Is there a metric separable space $X$ with the following properties: $|X|\geq\...
Alexander Osipov's user avatar
7 votes
1 answer
185 views

Existence of Borel uniformization for coanalytic set with non-$K_\sigma$ sections

Suppose that $X$ is a Polish (or standard Borel) space and $\omega^\omega$ is the Baire space of all natural number sequences. My question is: If $A\subseteq X\times \omega^\omega$ is a coanalytic set ...
Iian Smythe's user avatar
  • 3,115
11 votes
0 answers
172 views

Can the nowhere dense sets be more complicated than the meager sets?

Suppose $X$ is a completely metrizable space with no isolated points. Let $\mathcal{ND}_X$ denote the ideal of nowhere dense subsets of $X$, and let $\mathcal{M}_X$ denote the ideal of meager subsets ...
Will Brian's user avatar
  • 18.5k
8 votes
1 answer
351 views

"Compactness length" of Baire space

Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton? In more ...
Noah Schweber's user avatar
4 votes
0 answers
155 views

Two other variants of Arhangel'skii's Problem

This question is a follow up to another question of mine, which turned out to be easy (for background on Arhangel'skii's Problem see Arhangel'skii's problem revisited). Recall that a space is ...
Santi Spadaro's user avatar
3 votes
0 answers
246 views

"weakly functorial resolution" of quasi-compact T_1 topological space by quasi-compact Hausdorff space

I have an arguably weird question: Let $X$ be a quasi-compact $T_1$ topological space, could there be a construction that takes such an $X$ as input and outputs a surjection $$X' \to X$$ with the ...
S. Li's user avatar
  • 619
5 votes
0 answers
131 views

Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?

Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
Tim Campion's user avatar
  • 63.9k
10 votes
0 answers
242 views

Arhangel'skii's problem revisited

One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is ...
Santi Spadaro's user avatar
9 votes
1 answer
593 views

Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?

We work in ZFC. Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not isomorphic to $\mathbb{R}$. A field $E$ is ...
Jakobian's user avatar
  • 1,201
15 votes
1 answer
480 views

Topology and pcf theory

$\DeclareMathOperator\pcf{pcf}$For simplicity say $\aleph_\omega$ is a strong limit. Let $A=\pcf\{\aleph_n:n\in\omega\}$. Then it follows from basic properties of pcf operation that $X\subseteq A\...
n901's user avatar
  • 667
8 votes
1 answer
211 views

Can totally inhomogeneous sets of reals coexist with determinacy?

A special case of a theorem of Brian Scott (from On the existence of totally inhomogeneous spaces) is that there is a size-continuum set $S\subset\mathbb{R}$ such that if $x,y\in S$ are distinct then $...
Noah Schweber's user avatar
2 votes
0 answers
92 views

Can this order relation, defined in terms of all topological spaces, be defined in terms of the reals alone?

Let $K$ be the operator monoid under composition of Kuratowski's $14$ set operators generated by topological closure $k$ and complement $c.$ Kuratowski's 1922 paper gives the poset diagram of the ...
mathematrucker's user avatar
5 votes
0 answers
177 views

Do closed subsets of the generalised Cantor space have an analogue of the perfect set property?

For a regular uncountable cardinal $\kappa$, consider $2^\kappa$ with the "less than box topology" (tree topology? Easton/Bounded support topology?) in which basic open sets are of the form $...
Calliope Ryan-Smith's user avatar
10 votes
0 answers
323 views

Determinacy coincidence at $\omega_1$: is CH needed?

This is a follow-up to the last part of an old MSE answer of mine. Briefly, an analogue at $\omega_1$ of Steel's equivalence between clopen and open determinacy can be proved assuming $\mathsf{CH}$, ...
Noah Schweber's user avatar
17 votes
1 answer
569 views

Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty?

(cross-posted from this math.SE question) It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to ...
Cla's user avatar
  • 775
2 votes
0 answers
156 views

Do Grothendieck topoi with enough points satisfy the fan theorem internally?

Fourman and Hylland proved in the 80s that all spatial topoi satisfy the full fan theorem internally, while there are examples of localic topoi that do not satisfy it. This leads one to conjecture a ...
saolof's user avatar
  • 1,947
9 votes
2 answers
540 views

Can you fit a $G_\delta$ set between these two sets?

Every subset of $\mathbb N \times \mathbb N$ can be viewed as a relation on $\mathbb N$. The set $\mathcal P(\mathbb N \times \mathbb N)$ of all relations on $\mathbb N$ has a natural topology with ...
Will Brian's user avatar
  • 18.5k
10 votes
1 answer
460 views

An incomplete characterisation of the Euclidean line?

We say that a metric space $(X, d)$ is a Banakh space if for every $\rho \in \mathbb{R}_{> 0}$ and every $x \in X$, there are $a,b \in X$ such that $\{y \in X \, \vert \, d(x, y) = \rho\} = \{a, b\}...
Luc Guyot's user avatar
  • 7,893
3 votes
1 answer
126 views

What are the names of the following classes of topological spaces?

The closure of any countable is compact. The closure of any countable is sequentially compact. The closure of any countable is pseudocompact. The closure of any countable is a metric compact set.
Alexander Osipov's user avatar
1 vote
0 answers
70 views

A ZFC example of a star-$K$-Menger space which is not star-$K$-Hurewicz

An open cover $\mathcal U$ of a space $X$ is said to be $\gamma$-cover if $\mathcal U$ is infinite and for each $x\in X$, the set $\{U\in\mathcal U : x\notin U\}$ is finite. A space $X$ is said to be ...
Nur Alam's user avatar
  • 505
6 votes
1 answer
260 views

A ZFC example of a Menger space which is not Scheepers

$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...
Nur Alam's user avatar
  • 505
5 votes
1 answer
311 views

Infinite tensor/Fubini product of ultrafilters

Given an infinite family $\{\mathcal{F}_{\lambda}$, $\lambda <\kappa\}$, $\kappa \geq \omega_0$, of (ultra)filters of a set $X$, how it is defined the infinite tensor/Fubini product $$\bigotimes_{\...
BTN's user avatar
  • 53
3 votes
0 answers
113 views

Hereditarily Lindelöf spaces with density continuum

Since there are L-spaces (provably in ZFC), under CH we have regular, hereditarily Lindelöf spaces with density continuum. However, I cannot find an example of such a space under not CH, nor a proof ...
GAW's user avatar
  • 31
3 votes
1 answer
135 views

Characterization of the Scheepers property by Scheepers game

$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...
Nur Alam's user avatar
  • 505
2 votes
0 answers
98 views

Closed images of linearly ordered spaces

Is there a description of the class of continuous closed images of linearly ordered spaces?
Smolin Vlad's user avatar
3 votes
0 answers
141 views

Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?

Which cardinal $\kappa\geq \omega_1$ is critical for the following property: Let $X\subset \mathbb R$ and $\kappa>|X|\geq \omega_1$. Then there is an uncountable family $\{X_{\alpha}\}$ such that $...
Alexander Osipov's user avatar
2 votes
0 answers
171 views

Is there a Lusin space $X$ such that ...?

Is there a Lusin space (in the sense Kunen) $X$ such that $X$ is Tychonoff; $X$ is a $\gamma$-space ? Note that if $X$ is metrizable and a $\gamma$-space then it is not Lusin. In mathematics, a ...
Alexander Osipov's user avatar
5 votes
1 answer
370 views

Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?

Recall that $\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the sfip which has no infinite pseudo-intersection $\}$. The cardinal $\mathfrak{q}_0$ defined as the smallest ...
Alexander Osipov's user avatar
14 votes
1 answer
272 views

Is there a countably infinite closed interval in the lattice of topologies?

Is there an interval of the form $[\sigma,\tau]$ in the lattice of topologies on some set $X$ such that $|[\sigma,\tau]| = \aleph_0$? In other words, do there exist two topologies $\sigma$ and $\tau$ ...
Will Brian's user avatar
  • 18.5k
14 votes
1 answer
581 views

How “disconnected” can a continuum be?

A continuum is a compact connected metrizable topological space. Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint ...
Alessandro Codenotti's user avatar
3 votes
0 answers
167 views

What is the name of the class of topological spaces with the following property ....?

What is the name of the class of topological spaces with the following property $P$ ? $X\in P$ iff for any open set $W$ in $X$ and any point $x\in \overline{W}\setminus W$ there is an open set $V$ ...
Alexander Osipov's user avatar
4 votes
1 answer
94 views

Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space?

A topological space $X$ is called a $\sigma$-space if every $F_{\sigma}$-subset of $X$ is $G_{\delta}$. A topological space $X$ is called a $Q$-space if any subset of $X$ is $F_{\sigma}$. Definition. ...
Alexander Osipov's user avatar
4 votes
0 answers
127 views

An uncountable Baire γ-space without an isolated point exists?

An open cover $U$ of a space $X$ is: • an $\omega$-cover if $X$ does not belong to $U$ and every finite subset of $X$ is contained in a member of $U$. • a $\gamma$-cover if it is infinite and each $x\...
Alexander Osipov's user avatar
6 votes
0 answers
255 views

Every Polish space is the image of the Baire space by a continuous and closed map, reference

The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501) Every Polish space (i.e. every separable ...
Lorenzo's user avatar
  • 2,286
5 votes
0 answers
231 views

Does Tychonov's theorem directly imply Zorn's lemma?

This question was formerly posted on MSE https://math.stackexchange.com/questions/4578923/ without getting an answer. I know that Tychonov's theorem, Zorn's lemma, the axiom of choice, the well-...
Jochen Wengenroth's user avatar
5 votes
1 answer
217 views

How many pairwise non-homeomorphic non-empty closed subsets of the Cantor set are there? [duplicate]

My question is more or less related to basic set theory. But I don't know even that. Apologies if I added the wrong tags. Motivation: How many non-compact (planar) surfaces are there upto ...
Random's user avatar
  • 1,097
7 votes
1 answer
429 views

$\Sigma_*$-product is not $\sigma$-countably compact

In Arhangel'skii's book "Topological function spaces" there is a part where the author uses that, if $\kappa>\omega$ is a cardinal number, then the space $$\Sigma_*(\kappa):=\left\{x\in \...
Peluso's user avatar
  • 674
4 votes
0 answers
136 views

Is there a condensation of a closed subset of $\kappa^\omega$ onto $\kappa^\omega\setminus A$ …?

Let $\aleph_1\le\kappa<c$ and $A\subset \kappa^{\omega}$ such that $\lvert A\rvert\le\kappa$. Is there a condensation (i.e. a bijective continuous mapping) of a closed subset of $\kappa^\omega$ ...
Alexander Osipov's user avatar
6 votes
1 answer
191 views

Steinhaus number of a group

$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$. Let $\mathcal A_X$ be the family of ...
Taras Banakh's user avatar
  • 41.9k
4 votes
0 answers
142 views

Consistency of a strange (choice-wise) set of reals, pt. 2

This is a follow-up on this question. Consider a set $X\subseteq \mathbb{R}$ such that $X$ is not separable wrt its subspace topology Every countable family of non-empty pairwise disjoint subsets of $...
Lorenzo's user avatar
  • 2,286
0 votes
1 answer
176 views

What does mean by "$\omega +1$ is convergent sequence"? [closed]

Let $X=\omega +1$ be convergent sequence. Then what does mean by "$X$ is convergent sequence"?
Nur Alam's user avatar
  • 505
7 votes
0 answers
138 views

The smallest cardinality of a cover of a group by algebraic sets

$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
Taras Banakh's user avatar
  • 41.9k
7 votes
2 answers
722 views

Consistency of a strange (choice-wise) set of reals

Consider a set $X\subseteq \mathbb{R}$ such that $X$ is not separable wrt its subspace topology For all $r\in\mathbb{R}$ there exists a sequence $(x_n)_{n\in\omega} \subset X$ converging to $r$ In a ...
Lorenzo's user avatar
  • 2,286

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