Definition:Partition Topology
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Definition
Let $S$ be a set.
Let $\mathcal P$ 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 $\mathcal P$
Then $\tau$ is a partition topology on $S$, and $\left({S, \tau}\right)$ is a partition (topological) space.
The partition $\mathcal P$ is the basis of $\left({S, \tau}\right)$.
Also see
The above statements are proved in:
- Results about partition topologies can be found here.
Sources
- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 5$