Tietze extension theorem

The Tietze extension theorem states that every continuous real-valued function on a closed subset of a normal topological space extends continuously to the whole space.

Urysohn lemma: In a normal space, disjoint closed sets can be separated by a continuous function taking values in the unit interval. It supplies the separating functions used to construct extensions of bounded functions.
Topological space: A set equipped with a topology, a collection of subsets designated as open and closed under standard operations. The theorem concerns continuous functions and closed subsets within this general setting.
Tietze extension property: The Tietze extension property is the ability to extend continuous real-valued functions from closed subsets to an entire space. It names the topological property characterized by the theorem.
Tietze extension theorem for normal spaces: The normal-space formulation extends real-valued continuous functions from closed subsets of normal spaces. This is the theorem’s standard setting, with normality as its central separation hypothesis.
Uniform convergence: A sequence of functions converges uniformly when one bound on its error works at every point of the domain. The successive corrections converge uniformly, preserving continuity of their sum.
Closed set: A subset of a topological space is closed when its complement is open. The function is initially defined on a closed subset, a key condition of the theorem.
Continuous extension: A continuous extension agrees with a given function on its original domain and is defined continuously on a larger domain. The theorem guarantees such extensions when the subset is closed and the space is normal.
Urysohn metrization theorem: Urysohn’s metrization theorem gives conditions under which a topological space admits a compatible metric. It is another major theorem about normal spaces, but concerns metrization rather than function extension.
Truncation: Truncation replaces values outside a chosen interval by the nearest endpoint of that interval. Bounding the original function reduces the extension construction to controlled corrections.
Bounded function: A real-valued function is bounded if its values lie within some finite interval. The bounded form is the core extension result used to handle arbitrary real-valued functions.

Linked from 5 pages