Separation Axioms

Abstract

In general, the properties of topological spaces are quite different from those of metric spaces, so some additional restrictions are often imposed on the topology of a space in order to bring its properties closer to those of metric spaces. The reader might be aware of the fact that two distinct points or disjoint closed sets in a metric space can be separated by disjoint open sets. There are more than ten such “separation axioms” for topological spaces, traditionally denoted by $T_0,T_1,\ldots$. These specify the degree to which points or closed sets may be separated. The axioms $T_0$, $T_1$, and $T_2$ have already been studied in Chap. 4. In the present chapter, we shall treat the axioms $T_3$, $T_{3\frac{1}{2}}$, and $T_4$ only. Sect. 8.1 concerns with the axiom $T_3$ which stipulates the separation of points and closed sets by disjoint open sets. Sect. 8.2 is devoted to the axiom $T_4$ which specifies that every pair of disjoint closed sets have disjoint nbds. For spaces satisfying this axiom, we shall prove several important theorems of topology such as Urysohn’s Lemma, Tietze Extension Theorem, and Urysohn Metrization Theorem. The other axiom is stronger than $T_3$-axiom and weaker than $T_4$-axiom for most of the spaces. This requires separation of points and closed sets by real-valued continuous functions, and its study is the object of Sect. 8.3. It will be seen that the spaces satisfying axioms $T_1$ and $T_{3\frac{1}{2}}$ can be embedded into compact Hausdorff spaces; so some interesting theorems can be proved for them.