# Definition:Degenerate Connected Set

## Definition

Let $T = \struct {S, \tau}$ 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.

## Motivation

Despite the result Empty Set Satisfies Topology Axiomsâ€Ž, 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.