Quasicomponent is not necessarily Component

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
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