I have the following proof, but I don't understand one of the steps: Theorem 4.4. A Boolean algebra is complete iff its Stone space is exlremally disconnected. Proof. Identify the given ...
The wikipedia article on ordinal spaces claims that they are not extremally disconnected: However, they are not extremally disconnected in general (there is an open set, namely $\omega$, whose ...
Let $X$ be a topological space and let $\Gamma \mathcal O(X)$ be it's boolean algebra of clopen subsets. For compact totally disconnected space, show that $\Gamma \mathcal O(X)$ is complete (as a ...
An open subset U of a space X is regular if it equals the interior of its closure, as we learn from the Wikipedia glossary of topology. Furthermore, the regular open subsets of a space (any space) ...
In "Introduction to Boolean Algebras" the authors introduce a symbol for the complement of the closure of P, where P is a set in a topological space (Ch. 9, p. 60). This is in the context of ...
Let $B$ be a Stone space (compact, Hausdorff, and totally disconnected). Then I am basically certain (because of Stone's representation theorem) that if $a, b \in B$ are two distinct points in $B$, ...
What are the open sets of a Stone space of a Boolean algebra B? According to Wikipedia, they are the ultrafilters on B that contain an element of B. However, this cannot be the case since an ...