Please see Convention:Hausdorffness assumption
Equivalent definitions in tabular format
Note that under some conventions, the definition below is taken as the definition of normal space. However, under the convention followed in this wiki, the definition of normal space includes the assumption that points are closed, forcing the space to be a T1 space (and further also a Hausdorff space, based on the definition). The definition given here is obtained when we relax the requirement of points being closed. Note that this definition includes the normal spaces where all points are closed, but also includes some other spaces.
|No.||Shorthand||A topological space is said to be normal(-minus-Hausdorff) if ...||A topological space is said to be normal(-minus-Hausdorff) if ...|
|1||separation of disjoint closed subsets by open subsets||given any two disjoint closed subsets in the topological space, there are disjoint open sets containing them.||given any two closed subsets such that , there exist disjoint open subsets of such that , and .|
|2||separation of disjoint closed subsets by continuous functions||given any two disjoint closed subsets, there is a continuous function taking the value at one closed set and 1 at the other.||for any two closed subsets , such that , there exists a continuous map (to the closed unit interval) such that and .|
|3||point-finite open cover has shrinking||every point-finite open cover possesses a shrinking.||for any point-finite open cover of , there exists a shrinking : the form an open cover and .|