Normal space

A topological space in which every pair of disjoint closed sets has disjoint open neighborhoods. Normality is a separation property used to construct continuous functions and prove extension theorems.

Topological space: A set equipped with a collection of subsets, called open sets, satisfying axioms for unions and finite intersections. Normality is a property of this underlying structure.
Urysohn lemma: A theorem stating that disjoint closed sets in a normal space can be separated by a continuous function into the unit interval. It converts normality directly into a continuous numerical distinction between closed sets.
Regular space: A topological space in which a point and a disjoint closed set have disjoint open neighborhoods, under a specified convention about T1. Regularity separates points from closed sets; normality separates two closed sets.
Normality (topology): The property that every pair of disjoint closed sets in a topological space has disjoint open neighborhoods. This is the separation property defining normal spaces.
Closed set: A subset whose complement is open in a given topological space. Normality tests pairs of disjoint sets of this kind.
Urysohn function: A continuous function into the unit interval taking prescribed values on two disjoint closed sets. Such functions witness the separation guaranteed by the Urysohn lemma.
Completely normal space: A topological space in which every pair of separated sets has disjoint open neighborhoods. This stronger separation property extends normality beyond closed-set pairs.
Partition of unity: A family of continuous functions whose values sum to one and whose supports are locally finite. Normality helps construct partitions of unity when combined with paracompactness.
Open set: A member of a topology, with unions and finite intersections of such sets also belonging to the topology. These are the neighborhoods that must separate closed sets.
Continuous function: A function between topological spaces for which preimages of open sets are open. Normality supports constructing continuous functions with controlled values on closed sets.

Linked from 8 pages