In a normal topological space, any two disjoint closed sets can be separated by a continuous real-valued function taking the value zero on one set and one on the other. Proved by Pavel Urysohn, it is a foundational tool for constructing continuous functions in general topology.
Facts
StatementA topological space is normal if and only if any two disjoint closed subsets of it can be separated by a continuous function into the unit interval, taking the value 0 on one subset and 1 on the other. 1 Classification
Statement FormCharacterization Theorem 1 Connections
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Proved By
Source Urysohn's Lemma (Wikipedia)
Sources
1. Urysohn's Lemma (Wikipedia)
Wikimedia Foundationlead paragraph, theorem statement sentence
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.
Proved By: Pavel Urysohn, Lead paragraph
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be
In Group: General Topology, Lead paragraph
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.
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