Open Set in Partition Topology is Component

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Theorem

Let $T = \left({S, \tau}\right)$ be a partition topological space.

Then each of its open sets are components of $T$.


Proof

Let the partition $\mathcal P$ be a basis of $T$.

From Open Set in Partition Topology is also Closed, open sets are in fact clopen.

So the elements of $\mathcal P$ are clopen.

The result follows from the definition of components.

$\blacksquare$


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