Topological Space with One Quasicomponent is Connected

From ProofWiki
Jump to navigation Jump to search

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:

$\ds \bigcap K = S$

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

The result follows by definition of connected space.

$\blacksquare$


Sources