Ultraconnected implies normal

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This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., ultraconnected space) must also satisfy the second topological space property (i.e., normal space)
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Any ultraconnected space is a normal space.

Definitions used

Term Definitions used
ultraconnected space A space X is ultraconnected if any two non-empty closed subsets have non-empty intersection.
normal space A space X is normal if, given any two disjoint closed subsets A,B of X, there exist disjoint open subsets U,V of X such that A \subseteq U, B \subseteq V.


The proof is immediate from the observation that in an ultraconnected space, it is not possible to have disjoint non-empty closed subsets. Hence, given disjoint closed subsets, one of them must be empty, and we can use the empty space and whole space as the corresponding open subsets.