Iterations of resolvents and homogeneous cut-point spaces (Q887507)

If \((X,\tau)\) is a topological space in which to each point \(x\in X\) is assigned a topological space \(Y_x\) and a continuous mapping \(h_x : X \setminus \{x\} \to Y_x\), then the sets \((\{x\} \times V ) \cup\bigcup\{\{x^\prime\}\times Y_{x^\prime}: x^\prime\in U\cap h_x^{-1}[V]\}\), where \(x\in U\in \tau\) and \(V\) is open in \(Y_x\), are a base for a topology on the set \(R(X, Y_x, h_x) =\bigcup\{\{x\} \times Y_x : x \in X\}\). The space \(R(X, Y_x, h_x)\) (or simply \(R(X)\)) is the \textsl{resolvent of} \(X\) (at each point \(x \), by means of the mapping \(h_x\)). The mapping \(\pi : R(X) \to X\) defined by \(\pi(x,y)=x\) is called the \textsl{resolvent mapping}. The process of constructing resolvents can be iterated: Denote \(R_0 = X\), \(R_1 = R(X)\), and let \(\pi^{n+1}_n : R_{n+1} \to R_n\) be the resolvent mappings, where \(R_{n+1} = R(R_n)\) for each \(n\in\omega\). The limit \(R_\omega\) of the resulting inverse spectrum is the \(\omega\)-iteration of the resolvents. A connected space \(X\) is a \textsl{cut-point space of order} \(n\) if \(X\setminus\{x\}\) has exactly \(n\) components for each \(x\in X\); homogeneous cut-point spaces of order 3 were constructed by \textit{L. R. Ford jun.} [Trans. Am. Math. Soc. 77, 490--497 (1954; Zbl 0058.17302)] and \textit{D. Daniel} and \textit{W. S. Mahavier} [Int. J. Math. Math. Sci. 2007, Article ID 10679, 8 p. (2007; Zbl 1145.54013)]. In the paper under review, homogeneous cut-point spaces of arbitrary order are constructed using iterations of resolvents with constant mappings. Additionally, it is proved that such spaces are not algebraically homogeneous, that is to say, they are not the spaces of left cosets of a topological group.