Quasicomponent is not necessarily Component
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $Q$ be a quasicomponent of $T$.
Then it is not necessarily the case that $C$ is also a component of $T$.
Proof
From Component of Point is not always Intersection of its Clopen Sets, the set intersection of the clopen sets containing a point $x$ may not always be contained in the component of $x$.
The result follows from Quasicomponent is Intersection of Clopen Sets.
$\blacksquare$
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