All Questions
Tagged with gn.general-topology boolean-algebras
11 questions
21
votes
1
answer
1k
views
Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?
Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra.
Moreover, a morphism of commutative von Neumann algebras induces
a continuous morphism of the corresponding ...
- 37.8k
16
votes
2
answers
2k
views
Is Stone-Čech compactification of 0-dimensional space also 0-dimensional?
What is an example of a 0-dimensional locally compact Hausdorff space $X$ for which the Stone-Čech compactification $\beta(X)$ is not 0-dimensional?
It is known that if $X$ is a 0-dimensional locally ...
- 1,704
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
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
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
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
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
5
votes
1
answer
254
views
Boolean algebra of ambiguous Borel class
Suppose $X$, $Y$ are uncountable compact metric spaces and $\Delta^0_\xi(X)$, $\Delta^0_\xi(Y)$ ($2\le\xi\le\omega_1$) are the Boolean algebras of Borel sets of ambiguous class $\xi$. So for $\xi=2$ ...
- 1,704
3
votes
1
answer
906
views
Finitely additive measures on Boolean algebras of regular open subsets: Is there a relationship with Borel measures? A theory of integration?
Let $\mathcal{X}$ be a topological space. An open subset $\mathcal{R}\subseteq\mathcal{X}$ is regular if it is the interior of its own closure. The intersection of two regular open sets is regular. ...
- 577
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
2
votes
2
answers
588
views
What to call a continuous function with preimage preserving nowhere-density?
Currently I am reading some basic literature on descriptive set theory and boolean algebras. And this comes out a lot, for example in results like:
Let $X$ and $Y$ be topological spaces, and $f:X \to ...