On cut points properties of subclasses of connected spaces (Q2850632)

In this paper, the authors first establish the result that if a connected space \(X\) has a closed \(R(i)\) subset \(H\) such that there is no proper regular closed, connected subset of \(X\) containing \(H\), then there is no proper connected subset of \(X\) containing all non-cut points. Using this result, they show that a connected space \(X\) has at most two non-cut points and a closed \(R(i)\) subset \(H\) such that there is no proper regular closed, connected subset of \(X\) containing \(H\) if and only if \(X\) is a connected ordered topological space (COTS) with endpoints. Moreover, if such a space is locally connected, then it is compact.NEWLINENEWLINEThe authors also prove that, in an \(R(i)\) connected space \(X\), (1) every component of the complement of a cut point contains a non-cut point of \(X\) and (2) if the removal of every two-point disconnected set leaves \(X\) disconnected, then, for \(a, b \in X\) and a separation set \(H\) of \(X \setminus \{a, b\}\), \(H \cup \{a, b\}\) is a \(T_{\frac{1}{2}}\) \(R(i)\) COTS with endpoints.NEWLINENEWLINEFinally, the authors establish some additional characterizations of a COTS with endpoints and some characterizations of the closed unit interval.