Urysohn's lemma is a fundamental result in topology that plays a crucial role in the proof of several important theorems. It is particularly significant in the context of normal spaces, where it allows for the construction of continuous functions that can separate any two disjoint closed sets. This property is not only essential for understanding the structure of topological spaces but also forms the basis for proving other theorems such as the Urysohn's metrisation theorem, which states that any normal space is metrizable under certain conditions.
The lemma's role in proofs is highlighted by its application in demonstrating the existence of partitions of unity, which are vital in differential geometry and algebraic topology, and in establishing the Tietze extension theorem. These applications showcase Urysohn's lemma as a cornerstone in bridging the gap between abstract topological properties and more concrete, metric-based arguments.
For a deeper understanding, one can refer to detailed explanations and proofs available in resources like Wikipedia's article on Urysohn's lemma [1], or educational content such as the video series on the topic [5][6]. These resources provide comprehensive insights into how Urysohn's lemma is applied in proofs and its broader implications in the field of topology.