Open Set in Partition Topology is Component
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ be a partition topological space.
Then each of its open sets are components of $T$.
Proof
Let the partition $\PP$ be a basis of $T$.
From Open Set in Partition Topology is also Closed, open sets are in fact clopen.
So the elements of $\PP$ are clopen.
The result follows from the definition of components.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $5$. Partition Topology: $1$