Definition:Degenerate Connected Set

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$ be a subset of $T$.


$H$ is a degenerate connected set of $T$ if and only if it is a connected set of $T$ containing exactly one element.


Non-Degenerate Connected Set

A non-degenerate connected set of $T$ is a connected set of $T$ containing more than one element.


Degenerate Connected Space

When $H = S$ itself, the entire space can be referred to in this way:


$T$ is a degenerate connected space if and only if it contains 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.