All Questions
15 questions
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 ...
- 63.9k
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 ...
- 38.6k
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 $\hat{\Phi}$ ...
- 63.9k
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\...
- 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{...
- 1,112
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?
- 48.9k
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;
...
- 207
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 ...
- 3,115
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 ...
- 263
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 $X$ ...
- 1,112
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 ...
- 774
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 ...
- 1,151
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 ...
- 387
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 ...
- 55
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 $\...
- 596