Cut-Points with Separations in Semi and Pre-Connected Topological Spaces

Abstract

This paper develops a corrected and strengthened theory of cut-points in semi-connected and pre-connected topological spaces. Beginning with semi-open, pre-open, and alpha-open sets, explicit formulas for semi-closure and pre-closure are established and the standard identity αO(X) = SO(X) ∩ PO(X) is proved. A key structural result shows that, for the Njåstad notion of alpha-openness used here, alpha-connectedness is equivalent to ordinary connectedness. Consequently, classical and alpha-cut-points coincide. This places the genuinely different behavior at the semi- and pre-levels. If X is semi-connected (respectively, pre-connected), every classical cut-point is a semi-cut-point (respectively, pre-cut-point), but the converses fail. Two explicit finite spaces show that semi-cut-points and pre-cut-points are independent even when the ambient space is both semi-connected and pre-connected. The paper also corrects a subspace computation occurring in a standard three-point illustration and identifies a limitation of direct component-type generalizations: semi-openness is not hereditary to arbitrary subspaces, so a semi-component need not lie in one side of a semi-separation without an additional hypothesis. The resulting framework separates valid transfer principles from statements requiring extra assumptions and situates the theory within recent work on generalized openness, pretopological connectivity, and graph-associated topologies