What are the properties \(T_0\)–\(T_4\)? What other names are there for \(T_2\)–\(T_4\)?
Solution
Using the standard conventions:
- \(T_0\): for any distinct \(x,y\), at least one has an open neighborhood not containing the other.
- \(T_1\): for any distinct \(x,y\), each has an open neighborhood not containing the other; equivalently, every singleton is closed.
- \(T_2\): distinct points have disjoint open neighborhoods. This is the Hausdorff axiom.
- \(T_3\): \(T_1\) and regular: if \(x\notin F\) with \(F\) closed, then \(x\) and \(F\) have disjoint open neighborhoods.
- \(T_4\): \(T_1\) and normal: disjoint closed sets have disjoint open neighborhoods.
Thus \(T_2,T_3,T_4\) are commonly called Hausdorff, regular, and normal, respectively, with the understanding that the \(T_1\) condition is included in the latter two names under this convention.