Quotient Topology of Partition Topology is Discrete Space
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Theorem
Let $\PP$ be a partition of a set $S$.
Let $T = \struct {S, \tau}$ be the partition space formed from $\PP$.
Let $S / \PP$ be the quotient set of $S$ by $\PP$.
Then the quotient topology $\tau_{S / \PP}$ is a discrete topology.
Proof
Let $\BB$ be the set defined as:
- $\BB = \set {\set A: A \in S / \PP}$
From Basis for Partition Topology, $\BB$ forms a basis for a partition space on $S$.
From Basis for Discrete Topology, $\BB$ forms a basis for the discrete topology on $S / \PP$.
Hence the result, by definition of quotient topology.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $5$. Partition Topology: $1$