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16 votes
1 answer
607 views

The dominating number $\mathfrak{d}$ and convergent sequences

All spaces considered below are compact Hausdorff. If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...
Damian Sobota's user avatar
14 votes
2 answers
502 views

Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?

Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder. The map $j:n\...
YCor's user avatar
  • 63.9k
11 votes
1 answer
2k views

Is every complete Boolean algebra isomorphic to the quotient of a powerset algebra?

Is every complete Boolean algebra isomorphic to a quotient, as a Boolean algebra, of some powerset algebra $\wp(X)$? It is not true for arbitrary Boolean algebras, see the comments, or see my MathSE ...
Thomas's user avatar
  • 263
10 votes
1 answer
354 views

Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?

Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures. Consider the endomorphism $\hat{\Phi}$ ...
YCor's user avatar
  • 63.9k
8 votes
4 answers
714 views

Are there $2^{\aleph_0}$ pairwise non-isomorphic Boolean algebra structures on $\omega$?

Is there a collection of $2^{\aleph_0}$ pairwise non-isomorphic countable Boolean algebras? Equivalently, are there $2^{\aleph_0}$ pairwise non-homeomorphic closed subsets in the Cantor space?
Dominic van der Zypen's user avatar
8 votes
2 answers
289 views

Does $\aleph_0$-density of regular open algebra entail existence of countable basis?

Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as Boolean algebra) be $\aleph_0$. Does $X$ ...
Rafał Gruszczyński's user avatar
8 votes
1 answer
365 views

Counting copies of a BA within a BA: arbitrarily many vs infinitely many

Informally, I am wondering if a Boolean algebra $\mathcal{B}$ contains infinitely many disjoint copies of a Boolean algebra $\mathcal{A}$ whenever it contains arbitrarily many disjoint copies of $\...
Asher M. Kach's user avatar
7 votes
1 answer
397 views

A set theoretic question arising from trying to understand a sheaf cohomology question

I'm trying to understand the footnote to Example 5.3 in Wiegand - Sheaf cohomology of locally compact totally disconnected spaces which is about constructing a locally compact Hausdorff and totally ...
Benjamin Steinberg's user avatar
6 votes
2 answers
482 views

Complete atomless Boolean algebras with abelian automorphism group

Is there any example of a complete atomless Boolean algebra with a non-trivial abelian automorphism group? This is equivalent, by Stone duality, to asking for an extremally disconnected compact ...
Iian Smythe's user avatar
  • 3,115
5 votes
2 answers
314 views

Self-homeomorphism of Stone-Čech boundary with an isolated fixed point

$\DeclareMathOperator\bso{\beta^*\!\omega}\DeclareMathOperator\Homeo{Homeo}$Let $\bso$ be the complement of the countable discrete space $\omega$ in its Stone-Čech compactification $\beta\omega$ (some ...
YCor's user avatar
  • 63.9k
5 votes
1 answer
371 views

A problem of non-emptiness of intersections of certain chains of regular open sets

Let $X$ be a topological space and $\mathrm{RO}(X)$ its complete boolean algebra of regular opens. Define well inside relation: $$U\prec V\iff\overline{U}\subseteq V.$$ Let $\mathcal C\subseteq\mathrm{...
Rafał Gruszczyński's user avatar
3 votes
1 answer
436 views

Stone topological Boolean algebras

I am looking for an initial reference for a theorem which is known, namely: Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological ...
Evgeny Kuznetsov's user avatar
3 votes
1 answer
164 views

Algebras with countable chains only

Is there an example of an uncountable Boolean algebra $B$ in which every chain is countable and such that $\ell_\infty$ embeds into the Banach space $C(\mbox{Stone }B)$? The latter requirement is not ...
Bojan Kwitek's user avatar
2 votes
1 answer
220 views

Extending BAs to weakly countably distributive algebras.

Suppose $\mathscr{A}$ is a complete Boolean algebra (assume it is c.c.c. if you wish). Is there any, say, canonical embedding $\mathscr{A}\subseteq \mathscr{B}$ into a complete Boolean algebra which ...
TomK's user avatar
  • 55
2 votes
0 answers
240 views

3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators

During my studies, I came across several different Stone spaces, e.g.: (i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators; ...
LJGC's user avatar
  • 207