Singleton is Connected in Topological Space

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

Then the singleton $\set{x}$ is connected.

Proof

Let $A = \set{x}$.

From definition $3$ of a connected set, $A$ is connected in $T$ if and only if the subspace $\struct {A, \tau_A}$ is a connected space.

From Topology on Singleton is Indiscrete Topology, $\tau_A$ is the indiscrete topology.

From Indiscrete Space is Connected, $\struct {A, \tau_A}$ is a connected space.

$\blacksquare$