Topological Space with One Quasicomponent is Connected
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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
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness