Real Analysis Quals - Helpful Information

2.1. Basic properties: separation, countability, separability, compactness, connectedness. A space X is Tychonoff if for u, v ∈ X, there exists a neighborhood of u not containing v (and by symmetry a neighborhood of v not containing u). It is Hausdorff if every two points can be separated by disjoint neighborhoods. It is regular if it is Tychonoff and every closed set and point not in the set can be separated by disjoint neighborhoods. It is normal if it is Tychonoff and every two disjoint closed sets can be separated by disjoint neighborhoods. To be Tychonoff precisely means that every singleton set is closed. Every metric space is normal (easy to see using distance functions). If a space is Tychonoff, it is normal if and only if for U a neighborhood of a closed set F there is a neighborhood O such that