A topological space is normal when every two disjoint closed subsets have disjoint open neighbourhoods.
If is closed, is open, and in a normal topological space, separate from the closed complement by disjoint open sets . Then . In particular, first separate two disjoint closed sets by disjoint open sets and then shrink each neighborhood this way: their closures are disjoint. No additional metric or compactness assumption is needed.
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.
For a real function in the bounded continuous functions of norm on a closed subset of a metric space, separate its closed level sets and by . The distance to a set functions make continuous with the displayed bounds. Empty level sets are handled by the constants , or . Iterating on the residual yields a uniformly convergent continuous extension with no increase in the supremum norm.
A closed subset of a normal space is a countable intersection of open sets exactly when it is the zero set of some continuous function .
Articles by others on the same topic
In topology, a normal space is a specific type of topological space that satisfies certain separation properties. A topological space \( X \) is called **normal** if it meets the following criteria: 1. **It is a T1 space**: This means that for any two distinct points in the space, there exist open sets that contain one point but not the other. In other words, points can be separated by neighborhoods.