Urysohn's lemma

In a normal topological space, any two disjoint closed sets can be separated by a continuous function into the unit interval, taking values 0 and 1 on the respective sets.

Normal space: A topological space in which disjoint closed sets have disjoint open neighborhoods. This separation property supplies the open sets used to build the function.
Topological space: A set equipped with a collection of subsets designated as open and satisfying the topology axioms. The lemma is stated for spaces whose open sets determine continuity and separation.
Tietze extension theorem: A theorem extending bounded continuous real-valued functions from closed subsets of normal spaces. Its proof builds extensions from Urysohn functions that separate closed sets.
Completely regular space: A topological space where points and disjoint closed sets can be separated by a continuous real-valued function. Complete regularity separates points from closed sets, unlike normality's closed-set pair condition.
Dyadic rational: A rational number whose denominator is a power of two. Values at dyadic rationals index nested open sets in the standard construction.
Closed set: A subset whose complement is open. The two sets separated by the lemma must be closed.
Stone–Čech compactification: The largest compact Hausdorff compactification of a completely regular space. Urysohn functions help construct the continuous maps used to characterize this compactification.
Tychonoff space: A completely regular Hausdorff topological space. The lemma's normality hypothesis is stronger than the separation assumptions defining this class.
Nested neighborhoods: A sequence of neighborhoods arranged so that each lies inside the next. The construction recursively inserts neighborhoods between previously chosen closed and open sets.
Disjoint sets: Sets whose intersection is empty. Disjointness is the condition that permits distinct endpoint values.

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