Definition:Degenerate Connected Set

Definition
Let $T$ be a topological space.

A degenerate connected set of $T$ is a connected set of $T$ containing exactly one element.

Note
Despite the result Empty Set Satisfies Topology Axioms‎, the underlying set of a topological space is generally stipulated to be non-empty.

Hence the intuitive idea that a degenerate connected set may be allowed to contain no elements does not apply, because such a subset does not fulfil the conditions that define a topological space, and so cannot be defined as a connected space.