Urysohn Lemma

Urysohn’s Lemma is a fundamental result in topology, a branch of mathematics studying shapes and spaces. It states that in a well-behaved space (called normal space), given two separate closed sets, it’s possible to construct a continuous function that assigns values between 0 and 1, where one set maps entirely to 0, and the other to 1, with all points in between smoothly transitioning. This lemma shows how to create such "smooth" functions to distinguish different parts of a space, which is essential for understanding and manipulating complex geometric or topological structures.