Closure is Closed/Power Set
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It has been suggested that this page be renamed. In particular: Equivalence of Definitions of Closed Set under Closure Operator To discuss this page in more detail, feel free to use the talk page. |
Theorem
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\cl: \powerset S \to \powerset S$ be a closure operator.
Let $T \subseteq S$.
Then $\map \cl T$ is a closed set with respect to $\cl$.
Proof
By the definition of closure operator, $\cl$ is idempotent.
Therefore $\map \cl {\map \cl T} = \map \cl T$, so $\map \cl T$ is closed.
$\blacksquare$