D-2 - Higher Separation Axioms

Abstract

Three familiar separation axioms are defined by the requirement that certain pairs of disjoint sets have disjoint neighborhoods: a Hausdorff space (or a T2-space) is a space in which distinct points have disjoint neighborhoods; a regular space (or a T3-space) requires this for closed sets and points outside them; and in a normal space (or a T4-space), disjoint closed sets should have disjoint neighborhoods. According to Urysohn's Lemma in a normal space, disjoint closed sets are completely separated, that is, if A and B are closed and disjoint in the normal space X, then there is a continuous function. Normality of a product usually imposes extra structure on the factors. Collection wise, normality has been studied in connection with metrization theorems. Jones conjectured that normal Moore spaces are metrizable. This conjecture was called the Normal Moore Space Conjecture (NMSC). Moore spaces are first-countable and subparacompact and metrizable spaces are paracompact. Metrizable spaces are perfectly normal spaces; they are hereditarily normal but not perfectly normal space. Evidently, subspaces of a hereditarily normal space are hereditarily normal, and it is not difficult to show that subspaces of a perfectly normal space are also perfectly normal.