Topological Space with One Quasicomponent is Connected

Theorem
Let $T = \struct {S, \tau}$ be a topological space which has one quasicomponent.

Then $T$ is connected.

Proof
Let $x \in S$.

By hypothesis, the quasicomponent of $x$ is $S$ itself.

Thus by definition of quasicomponent:


 * $\forall y \in S: y \sim x$

where $\sim$ is the relation defined on $T$ as:


 * $x \sim y \iff T$ is connected between the two points $x$ and $y$

Let $K = \ds \bigcap_{x \mathop \in U} U: U$ is clopen in $T$.

By Quasicomponent is Intersection of Clopen Sets:
 * $\displaystyle \bigcap K = S$

Thus there is no non-empty clopen set of $T$ apart from $S$.

The result follows by definition of connected space.