\(H(i)\) connected topological spaces and cut points (Q1004044)

A topological space (no separation axiom is assumed) is an \textsl{\(H(i)\)-space} if every open cover has a finite dense subsystem (the term \textsl{\(\theta\)-compact space} has also been used for this concept). Generalizing results known for the class of compact Hausdorff spaces, the authors show that every connected \(H(i)\)-space has at least two non-cut points and that an \(H(i)\)-space with exactly two non-cut points is a COTS (a connected ordered topological space as defined by \textit{E. D. Khalimsky, R. D. Kopperman} and \textit{P. R. Meyer} [Topology Appl. 36, 1--17 (1990; Zbl 0709.54017)]), hence is a \(T_{1\over 2}\)-space with end-points.