Irreducible partitions and the construction of quasi-measures (Q2701695)

Let \(X\) be a connected, locally connected and compact Hausdorff space. A subset \(A\subset X\) is said to be solid if both \(A\) and \(X\setminus A\) are connected. An irreducible partition \({\mathcal P}= \{C_i\}_{1\leq i\leq n}\cup \{U_\lambda\}_{\lambda\in \Lambda}\) is a connection of disjoint solid sets such thatNEWLINENEWLINENEWLINE(1) Each \(C_i\) is closed and \(U_\lambda\) is open,NEWLINENEWLINENEWLINE(2) \(X= \bigcup^n_{i=1} C_i\cup \bigcup_{\lambda\in\Lambda} U_\lambda\),NEWLINENEWLINENEWLINE(3) If \(J\) is a proper subset of \(\{1,2,\dots, n\}\), then \(X\setminus \bigcup_{i\in J} C_i\) is connected.NEWLINENEWLINENEWLINELet \(g(X)\) be the maximum value of \(n-1\) for irreducible partitions \({\mathcal P}= \{C_i\}_{1\leq i\leq n}\cup \{U_\lambda\}_{\lambda\in \Lambda}\) of \(X\), and assume further the following: (a) \(g(X)= 1\), (b) If \(C\), \(C'\) are disjoint closed, solid, two-sided sets, then \(\{C,C'\}\) generates an irreducible partition of \(X\).NEWLINENEWLINENEWLINEThen the author gives an explicit construction of solid set functions (and, consequently, construction of quasi-measures) on a variety of such spaces \(X\), which is the main aim of this paper. He also notes that the examples presented here are the only quasi-measures known for spaces with \(g(X)= 1\) except for those that are images from spaces with \(g(X)= 0\).