Partition Topology is Topology
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Theorem
Let $S$ be a set.
Let $\PP$ be a partition of $S$.
Let $\tau$ be the set of subsets of $S$ defined as:
- $a \in \tau \iff a$ is the union of sets of $\PP$
Then $\tau$ is a topology on $S$.
Proof
From Basis for Partition Topology, we have that $\PP$ is a basis for the partition topology.
The result follows.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $5$. Partition Topology