R_1

Definition

• A topological space X is a R_1 space if any two topologically distinguishable points in X can be separated by neighbourhoods.

Property

• R_1 ⇒ mildly Hausdorff. [2]
• Every S-closed? R_1 space is extremally disconnected?.
• Every R_1 topological space is presober. [3]

Reference

1. a
2. Dontchev, J.; Popvassilev, S.; Stavrova, D., On the η-expansion topology for the co-semi-regularization and mildly Hausdorff spaces., (English summary), Acta Math. Hungar. 80 (1998), no. 1-2, 9-19.
3. -M.L. Colasante and D. Van der Zypen, Minimal R_1, minimal regular and minimal presober topologies, Revista Notas de Matemática, Vol.5(1),No. 275, 2009, pp.73-84