All Questions
Tagged with gn.general-topology boolean-algebras
48 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
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
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
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
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
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
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
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
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
6
votes
1
answer
678
views
Is it possible to define a closure operator in terms of partial ordering?
For boolean algebra, let's take Roman Sikorski's Boolean Algebras as our reference. After giving a set of axioms, he proves (p.9) that the join of A and B is the least element of the algebra such that ...
- 327
6
votes
1
answer
312
views
Can this ultrafilter convergence condition be expressed as a compactness condition?
Suppose that $X$ is a topological space. Let us say that an ultrafilter $\mathcal U$ on the Boolean algebra $C_X$ of clopen subsets of $X$ is partition-prime if whenever $X = \amalg_{i \in I} X_i$ is ...
- 373
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
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
5
votes
2
answers
257
views
Quotients of Cantor cubes onto spaces
Let $\lambda$ be an infinite cardinal. Consider the Cantor cube $\Delta_\lambda = \{0,1\}^\lambda$. It is a standard fact in topology that the topological weight (= minimal cardinality for a basis) of ...
- 387
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
5
votes
1
answer
167
views
(A kind of) Irreducibiliy of regular open convex sets in the Cartesian space
I am looking for a proof of the fact which is formulated at the bottom of this post. The property of regular convex sets which the fact expresses seems to be true to me, yet I have not been able to ...
- 1,112
4
votes
1
answer
203
views
Generalized limits in Boolean algebras
Let $\mathbb{B}$ be an infinite $\sigma$-complete Boolean algebra. By $\mathbb{B}^\omega$ we denote the countable product of $\mathbb{B}$ with the coordinate-wise operations. Let us call a ...
- 1,151
4
votes
1
answer
386
views
Functor from rings into compact Hausdorff spaces
There is an adjunction $\text{Bool}^{op} \leftrightarrow \text{Set}$ between boolean algebras and sets which sends a boolean algebra to the set of its prime ideals and a set $X$ to $[X, \mathbb{F}_2]_{...
user30211
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
3
votes
2
answers
384
views
ED compact $K$ such that $C(K)$ is not a dual Banach space
Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not hyperstonean). ...
- 55
3
votes
1
answer
161
views
Stone-Čech compactification of a Boolean subalgebra of $\{0,1\}^S$
Setup: Let $S$ be a set. Let $B$ be a Boolean subalgebra of $\{0,1\}^S$; i.e., just to be clear $B$ contains the constant $0$ and $1$ functions, and is stable under binary pointwise $\land$, $\lor$ ...
- 32.5k
3
votes
1
answer
129
views
perfect space without convergent long sequences
Is there a boolean space $X$ without isolated points with the property that no point $x\in X$ is the limit of a long sequence $(x_i)_{i\in I}$ from $X\setminus \lbrace x\rbrace $ ('long sequence' here ...
- 328
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
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
3
votes
0
answers
52
views
Does there exist a multi-valued "monotone" and "compact" map from a Boolean algebra to the "free" part of $\mathcal{P}(\kappa)$?
This is a follow-up to my previous question, which has a negative answer. Here is the most general version that I'm interested:
Does there exist a Boolean algebra $A$, an infinite cardinal $\kappa$, ...
- 2,830
3
votes
1
answer
216
views
Existence of a quasi-open (a.k.a semi-open) map into a Cantor cube
Recall that a topological space is extremally disconnected if the closure of any open set is open.
A continuous map is quasi-open if it maps nonempty open sets onto sets with nonempty interior. For ...
- 5,529
2
votes
1
answer
172
views
Separability of the Stone space of a free sigma-algebra
Let $X$ be the Stone space of the free $\sigma$-algebra $A$ on $\omega_1$ free generators.
Is $X$ separable (i.e. does $X$ contain a countable dense set)?
- 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 ...
2
votes
2
answers
249
views
What's "serialization" really called, and is there any theory surrounding it?
Define an operator $\mathop{\vec{\bigcup}}$ as follows:
Definition. Whenever $A$ is an $I$-indexed family of sets, where $I$ is a totally-ordered set, we have $$\mathop{\vec{\bigcup}}_{i \in I} A_i ...
- 3,793
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
2
votes
1
answer
185
views
Complete CCC Boolean algebras (or Stonean spaces)
I am interested in what is known about complete Boolean algebras $B$ with the countable chain condition (ccc), i.e., every disjoint set is countable. Let $K$ be the Stone space of $B$; the ...
- 388
2
votes
1
answer
149
views
Regular open Boolean algebras and homomorphism which does not preserve nearness of sets
I am looking for an example of topological spaces $\langle X_1,\mathscr{O}_1\rangle$ and $\langle X_2,\mathscr{O}_2\rangle$ such that
there is a homomorphism $h\colon\mathrm{r}\mathscr{O}_1\...
- 1,112
2
votes
1
answer
140
views
Is a Boolean algebra with an order continuous topology a measure algebra?
Assume that $B$ is a complete boolean algebra endowed with a Hausdorff topology, with respect to which all operations on $B$ are continuous, $0$ has a base of full sets (recall that $A\subset B$ is ...
- 5,529
2
votes
0
answers
96
views
Projective objects for compact po-spaces
Let us consider the following definition: a compact po-space is a pair $(X,\leq)$ where $X$ is a compact space and $\leq$ is an order, closed on $X^2$. Then, we can consider the category $KPoSp$ whose ...
- 21
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
0
answers
73
views
Dual equivalence for multioperators
This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...
- 1,017
1
vote
2
answers
235
views
Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?
Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be regular open if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A ...
- 48.9k
1
vote
2
answers
132
views
Description of atomless complete Boolean algebras with a countable $\pi$-base
Recall that a subset $A$ of a Boolean algebra $B$ is a $\pi$-base if for every $b>0$ there is $a\in A$ with $0<a\le b$. For example, the definition of atomicity says that atoms constitute a $\pi$...
- 5,529
1
vote
1
answer
119
views
Extremally disconnected rigid infinite Hausdorff compacta(?)
Question: does there exist an extremally disconnected infinite Hausdorff compact space $\ X\ $ such that the only homeomorphism
$\ h: X\to X\ $ is the identity homeomorphism
$\ h=\mathbb I_X:\ X\to X\...
- 4,786
1
vote
1
answer
114
views
Continuous surjection between spectra of commutative von Neumann algebras
Suppose that $V_1,V_2$ are two commutative von Neumann algebras and $V_1 \subset V_2$. Being in particular commutative $C^*$-algebras we have that $V_1 \cong C(X_1), V_2 \cong C(X_2)$ for some ...
- 9,330
1
vote
1
answer
176
views
Interval topology on complete Boolean algebras
Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
- 48.9k
1
vote
1
answer
250
views
Understanding Kelley's intersection number (Boolean algebras)
It is known that:
Theorem (Kelley, 1959). There exists a finite, strictly positive, finitely additive measure on a Boolean algebra $A$ if and only if $A^+$ is the union of a countable number of ...
- 11
1
vote
1
answer
137
views
Density and compactness of Boolean embeddings
Let A and B be Boolean algebras and $h:A\rightarrow B$ a
Boolean embedding.
If every element of $B$ can be expressed both as a join
of meets and as a meet of joins of elements in $h(A)$, then the ...
- 281
1
vote
1
answer
172
views
Ordering a subset of the clopens of a Stone space
Let $P$ be a countably infinite set of propositional variables and $\mathcal{L}_P$ be the propositional language generated from $P$ and the usual connectives $\wedge$, $\neg$, $\vee$. The set $\...
- 125
1
vote
1
answer
630
views
Is the boundary of an open set in a $\sigma$-space empty?
Recall that a Boolean space is a $\sigma$-space in case the closure of every open Borel set is open.
Let $\{B_i\}$ be a denumerable family of open-closed sets in a $\sigma$-space $X$. Then $\bigcup_i ...
- 175
0
votes
0
answers
41
views
Selectively countable Boolean algebras of sets (terminology)
I am interested in the name for the following property of a Boolean algebra $\mathcal A$ of subsets of a set $X$:
$(\star)$ for any sequence $(A_n)_{n\in\omega}$ of pairwise disjoint nonempty sets in $...
- 41.8k