Normal space

Please see Convention:Hausdorffness assumption

Definition

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 $X$ 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 $A,B\subseteq X$ such that $A\cap B=\varnothing$, there exist disjoint open subsets $U,V$ of $X$ such that $A\subseteq U,B\subseteq V$, and $U\cap V=\varnothing$.
2	separation of disjoint closed subsets by continuous functions	given any two disjoint closed subsets, there is a continuous function taking the value $0$ at one closed set and 1 at the other.	for any two closed subsets $A,B\subseteq X$, such that $A\cap B=\varnothing$, there exists a continuous map $f:X\to [0,1]$ (to the closed unit interval) such that $f(x)=0\ \forall x\in A$ and $f(x)=1\ \forall \ x\in B$.
3	point-finite open cover has shrinking	every point-finite open cover possesses a shrinking.	for any point-finite open cover $U_{i},i\in I$ of $X$, there exists a shrinking $V_{i},i\in I$: the $V_{i}$ form an open cover and ${\overline {V_{i}}}\subseteq U_{i}$.