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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 completeness of $B$ is equivalent to the fact that $K$ is Stonean (extremally disconnected compact Hausdorff).

Let $K_d$ be the closure of the isolated points of $K$ and denote the complement by $K_c$, so $K = K_d \sqcup K_c$; ccc implies that there are countably many isolated points. On the other hand, [Banach spaces of continuous functions, Theorem 5.1.3] (originally by Dixmier) shows that $K = K_h \sqcup K_a$ where $K_h$ is hyperstonean and $K_a$ has no nontrivial normal measures, and $K_a = K_m \sqcup K_n$ with $K_m$ containing a dense meagre subset and in $K_n$ every meagre subset is nowhere dense and the support of every measure is nowhere dense. Denote the measure algebra of the unit interval $\mathbb{I}$ (Lebesgue measurable sets modulo null sets) by $\Sigma$. By Maharan's Theorem $K_h = K_d \sqcup K_\Sigma$ where $K_\Sigma$ is either the Stone space of $\Sigma$, or empty. (All these notations were made up by me.) It follows that $$ K = K_d \sqcup K_\Sigma \sqcup K_m \sqcup K_n $$ The unique atomless separable complete Boolean algebra (see [Banach spaces of continuous functions, Theorem 1.7.11] or Description of atomless complete Boolean algebras with a countable $\pi$-base) has a Stone space homemorphic to $\mathbb{G}$, the Gleason cover of the unit interval $\mathbb{I}$, which has no normal measures, so if it exists, it is contained in $K_a = K_m \sqcup K_n$. In which part is it contained? ([Banach spaces of continuous functions, Example 5.1.4] does mention this fact about $\mathbb{G}$ but fails to answer this question.) What else is know about $K_a = K_m \sqcup K_n$ or equivalently, its Boolean algebra counterparts? Is some kind of characterization known?

EDIT: To elaborate on the Gleason part: let $B_c$ be the purely nonatomic part of $B$ and let $A := \{b \in B_c \colon [0,b] \text{ is separable}\}$. By ccc, $\sup A$ is a countable supremum and so $[0, \sup A]$ is also separable and anything disjoint with it is atomic or nonseparable. If $\sup A \not= 0$ then it corresponds to a clopen set in $K$ homeomorphic to $\mathbb{G}$.

(A Boolean algebra $B$ is separable if there exists a countable $D \subseteq B$ with $\forall b \in B \; \exists d \in D \; 0 < d \leq b$.)

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 completeness of $B$ is equivalent to the fact that $K$ is Stonean (extremally disconnected compact Hausdorff).

Let $K_d$ be the closure of the isolated points of $K$ and denote the complement by $K_c$, so $K = K_d \sqcup K_c$; ccc implies that there are countably many isolated points. On the other hand, [Banach spaces of continuous functions, Theorem 5.1.3] (originally by Dixmier) shows that $K = K_h \sqcup K_a$ where $K_h$ is hyperstonean and $K_a$ has no nontrivial normal measures, and $K_a = K_m \sqcup K_n$ with $K_m$ containing a dense meagre subset and in $K_n$ every meagre subset is nowhere dense and the support of every measure is nowhere dense. (All these notations were made up by me.) Denote the measure algebra of the unit interval $\require{enclose}\enclose{horizontalstrike}{\mathbb{I}}$ (Lebesgue measurable sets modulo null sets) by $\enclose{horizontalstrike}{\Sigma}$. By Maharan's Theorem $\enclose{horizontalstrike}{K_h = K_d \sqcup K_\Sigma}$ where $\enclose{horizontalstrike}{K_\Sigma}$ is either the Stone space of $\enclose{horizontalstrike}{\Sigma}$, or empty. The Boolean algebra corresponding to $K_h$ is classified by Maharan's Theorem (in particular, $K_d \subseteq K_h$). It follows that $$ K = K_h \sqcup K_m \sqcup K_n $$ The unique atomless separable complete Boolean algebra (see [Banach spaces of continuous functions, Theorem 1.7.11] or Description of atomless complete Boolean algebras with a countable $\pi$-base) has a Stone space homemorphic to $\mathbb{G}$, the Gleason cover of the unit interval $\mathbb{I}$, which has no normal measures, so if it exists, it is contained in $K_a = K_m \sqcup K_n$. In which part is it contained? ([Banach spaces of continuous functions, Example 5.1.4] does mention this fact about $\mathbb{G}$ but fails to answer this question.) What else is know about $K_a = K_m \sqcup K_n$ or equivalently, its Boolean algebra counterparts? Is some kind of characterization known?

EDIT: To elaborate on the Gleason part: let $B_c$ be the purely nonatomic part of $B$ and let $A := \{b \in B_c \colon [0,b] \text{ is separable}\}$. By ccc, $\sup A$ is a countable supremum and so $[0, \sup A]$ is also separable and anything disjoint with it is atomic or nonseparable. If $\sup A \not= 0$ then it corresponds to a clopen set in $K$ homeomorphic to $\mathbb{G}$.

(A Boolean algebra $B$ is separable if there exists a countable $D \subseteq B$ with $\forall b \in B \; \exists d \in D \; 0 < d \leq b$.)

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 completeness of $B$ is equivalent to the fact that $K$ is Stonean (extremally disconnected compact Hausdorff).

Let $K_d$ be the closure of the isolated points of $K$ and denote the complement by $K_c$, so $K = K_d \sqcup K_c$; ccc implies that there are countably many isolated points. On the other hand, [Banach spaces of continuous functions, Theorem 5.1.3] (originally by Dixmier) shows that $K = K_h \sqcup K_a$ where $K_h$ is hyperstonean and $K_a$ has no nontrivial normal measures, and $K_a = K_m \sqcup K_n$ with $K_m$ containing a dense meagre subset and in $K_n$ every meagre subset is nowhere dense and the support of every measure is nowhere dense. Denote the measure algebra of $[0,1]$ (Lebesgue measurable sets modulo null sets) by $\Sigma$. By Maharan's Theorem $K_h = K_d \sqcup K_\Sigma$ where $K_\Sigma$ is either the Stone space of $\Sigma$, or empty. (All these notations were made up by me.) It follows that $$ K = K_d \sqcup K_\Sigma \sqcup K_m \sqcup K_n $$ The unique atomless separable complete Boolean algebra (see [Banach spaces of continuous functions, Theorem 1.7.11] or Description of atomless complete Boolean algebras with a countable $\pi$-base) has a Stone space homemorphic to the Gleason cover of $[0,1]$ which has no normal measures, so if it exists, it is contained in $K_a = K_m \sqcup K_n$. In which part is it contained? ([Banach spaces of continuous functions, Example 5.1.4] does mention this fact about the Gleason cover but fails to answer this question.) What else is know about $K_a = K_m \sqcup K_n$ or equivalently, its Boolean algebra counterparts? Is some kind of characterization known?

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 completeness of $B$ is equivalent to the fact that $K$ is Stonean (extremally disconnected compact Hausdorff).

Let $K_d$ be the closure of the isolated points of $K$ and denote the complement by $K_c$, so $K = K_d \sqcup K_c$; ccc implies that there are countably many isolated points. On the other hand, [Banach spaces of continuous functions, Theorem 5.1.3] (originally by Dixmier) shows that $K = K_h \sqcup K_a$ where $K_h$ is hyperstonean and $K_a$ has no nontrivial normal measures, and $K_a = K_m \sqcup K_n$ with $K_m$ containing a dense meagre subset and in $K_n$ every meagre subset is nowhere dense and the support of every measure is nowhere dense. Denote the measure algebra of the unit interval $\mathbb{I}$ (Lebesgue measurable sets modulo null sets) by $\Sigma$. By Maharan's Theorem $K_h = K_d \sqcup K_\Sigma$ where $K_\Sigma$ is either the Stone space of $\Sigma$, or empty. (All these notations were made up by me.) It follows that $$ K = K_d \sqcup K_\Sigma \sqcup K_m \sqcup K_n $$ The unique atomless separable complete Boolean algebra (see [Banach spaces of continuous functions, Theorem 1.7.11] or Description of atomless complete Boolean algebras with a countable $\pi$-base) has a Stone space homemorphic to $\mathbb{G}$, the Gleason cover of the unit interval $\mathbb{I}$, which has no normal measures, so if it exists, it is contained in $K_a = K_m \sqcup K_n$. In which part is it contained? ([Banach spaces of continuous functions, Example 5.1.4] does mention this fact about $\mathbb{G}$ but fails to answer this question.) What else is know about $K_a = K_m \sqcup K_n$ or equivalently, its Boolean algebra counterparts? Is some kind of characterization known?

EDIT: To elaborate on the Gleason part: let $B_c$ be the purely nonatomic part of $B$ and let $A := \{b \in B_c \colon [0,b] \text{ is separable}\}$. By ccc, $\sup A$ is a countable supremum and so $[0, \sup A]$ is also separable and anything disjoint with it is atomic or nonseparable. If $\sup A \not= 0$ then it corresponds to a clopen set in $K$ homeomorphic to $\mathbb{G}$.

(A Boolean algebra $B$ is separable if there exists a countable $D \subseteq B$ with $\forall b \in B \; \exists d \in D \; 0 < d \leq b$.)

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 completeness of $B$ is equivalent to the fact that $K$ is Stonean (extremally disconnected compact Hausdorff).

Let $K_d$ be the closure of the isolated points of $K$ and denote the complement by $K_c$, so $K = K_d \sqcup K_c$; ccc implies that there are countably many isolated points. On the other hand, [Banach spaces of continuous functions, Theorem 5.1.3] (originally by Dixmier) shows that $K = K_h \sqcup K_a$ where $K_h$ is hyperstonean and $K_a$ has no nontrivial normal measures, and $K_a = K_m \sqcup K_n$ with $K_m$ containing a dense meagre subset and in $K_n$ every meagre subset is nowhere dense and the support of every measure is nowhere dense. Denote the measure algebra of $[0,1]$ (Lebesgue measurable sets modulo null sets) by $\Sigma$. By Maharan's Theorem $K_h = K_d \sqcup K_\Sigma$ where $K_\Sigma$ is either the Stone space of $\Sigma$, or empty. (All these notations were made up by me.) It follows that $$ K = K_d \sqcup K_\Sigma \sqcup K_m \sqcup K_n $$ The unique atomless separable Boolean algebra (see [Banach spaces of continuous functions, Theorem 1.7.11] or Description of atomless complete Boolean algebras with a countable $\pi$-base) has a Stone space homemorphic to the Gleason cover of $[0,1]$ which has no normal measures, so if it exists, it is contained in $K_a = K_m \sqcup K_n$. In which part is it contained? ([Banach spaces of continuous functions, Example 5.1.4] does mention this fact about the Gleason cover but fails to answer this question.) What else is know about $K_a = K_m \sqcup K_n$ or equivalently, its Boolean algebra counterparts? Is some kind of characterization known?

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 completeness of $B$ is equivalent to the fact that $K$ is Stonean (extremally disconnected compact Hausdorff).

Let $K_d$ be the closure of the isolated points of $K$ and denote the complement by $K_c$, so $K = K_d \sqcup K_c$; ccc implies that there are countably many isolated points. On the other hand, [Banach spaces of continuous functions, Theorem 5.1.3] (originally by Dixmier) shows that $K = K_h \sqcup K_a$ where $K_h$ is hyperstonean and $K_a$ has no nontrivial normal measures, and $K_a = K_m \sqcup K_n$ with $K_m$ containing a dense meagre subset and in $K_n$ every meagre subset is nowhere dense and the support of every measure is nowhere dense. Denote the measure algebra of $[0,1]$ (Lebesgue measurable sets modulo null sets) by $\Sigma$. By Maharan's Theorem $K_h = K_d \sqcup K_\Sigma$ where $K_\Sigma$ is either the Stone space of $\Sigma$, or empty. (All these notations were made up by me.) It follows that $$ K = K_d \sqcup K_\Sigma \sqcup K_m \sqcup K_n $$ The unique atomless separable complete Boolean algebra (see [Banach spaces of continuous functions, Theorem 1.7.11] or Description of atomless complete Boolean algebras with a countable $\pi$-base) has a Stone space homemorphic to the Gleason cover of $[0,1]$ which has no normal measures, so if it exists, it is contained in $K_a = K_m \sqcup K_n$. In which part is it contained? ([Banach spaces of continuous functions, Example 5.1.4] does mention this fact about the Gleason cover but fails to answer this question.) What else is know about $K_a = K_m \sqcup K_n$ or equivalently, its Boolean algebra counterparts? Is some kind of characterization known?