On minimal \(\pi\)-character of points in extremally disconnected compact spaces (Q1183638)

The refinement number \(r({\mathcal B})\) of a Boolean algebra \(\mathcal B\) is the minimal power of a set \(X\subset {\mathcal B}^ +\) such that for every \(a\in {\mathcal B}^ +\) there exists some \(x\in X\) such that either \(x\leq a\) or \(x\wedge a=0\). Similarly, \(r_{\text{fin}}({\mathcal B})=\min\{| X| : X\subset{\mathcal B}^ +\) and for every finite partition \(\mathcal P\) of \(\mathcal B\) there exists some \(p\in{\mathcal P}\) and \(x\in X\) such that \(x\leq p\}\). The definition of \(r_ \infty({\mathcal B})\) is obtained from the previous one by replacing ``finite partition'' by ``arbitrary partition''. Clearly, \(r({\mathcal B})\leq r_{\text{fin}}({\mathcal B})\leq r_{\infty}({\mathcal B})\). The authors obtain several deep and very interesting relationships between these cardinals and the \(\pi\)-weight and \(\pi\)-character in complete Boolean algebras. In particular, there is proved that if a Boolean algebra \(\mathcal B\) is homogeneous or complete, then the refinement number of \(\mathcal B\) is equal to the minimal \(\pi\)-character of an ultrafilter in \(\mathcal B\); in other words \(r(\mathcal B)=r_{\text{fin}}({\mathcal B})\). Also, the authors proved that if every cardinal \(\lambda\) of uncountable cofinality satisfies \(\lambda=\lambda^ \omega\), then for every complete Boolean algebra \(\mathcal B\), \(r_{\text{fin}}({\mathcal B})=r_ \infty({\mathcal B})\). However, the question of whether the inequality \(r_{\text{fin}}({\mathcal B})<r_ \infty({\mathcal B})\) can happen for a complete Boolean algebra, is left open. The authors apply their results concerning the refinement number to the following result: if \(X\) is a \(ccc\) extremally disconnected compact space such that the minimal character of a point in \(X\) equals the minimal \(\pi\)-weight of an open subset of \(X\) and is not greater than \(2^ \omega\), then \(X\) contains a discretely untouchable point, i.e. a point \(x\) such that for every countable discrete set \(N\subset X-\{x\}\), \(x\not\in \text{cl }N\). The existence of discretely untouchable points implies the famous Frolík's Theorem that every infinite compact extremally disconnected compact space is non-homogeneous.