Normal topological space

A topological space is normal when every two disjoint closed subsets have disjoint open neighbourhoods.

If $A,B$ are disjoint closed subsets of a normal space, there is a continuous function $f:X\to [0,1]$ with $f{|}_A=0$ and $f{|}_B=1$.

Every continuous real-valued function on a closed subspace of a normal space extends continuously to the whole space. If its values lie in a closed interval, the extension can be kept in that interval.

A closed subset $A$ of a normal space is a countable intersection of open sets exactly when it is the zero set of some continuous function $f:X\to [0,1]$.