Equivalence of Definitions of Closed Element
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $\cl$ be a closure operator on $S$.
Let $x \in S$.
The following definitions of the concept of Closed Element are equivalent:
Definition 1
The element $x$ is a closed element of $S$ (with respect to $\cl$) if and only if $x$ is a fixed point of $\cl$:
- $\map \cl x = x$
Definition 2
The element $x$ is a closed element of $S$ (with respect to $\cl$) if and only if $x$ is in the image of $\cl$:
- $x \in \Img \cl$
Proof
Let $\struct {S, \preceq}$ be an ordered set.
Let $\cl: S \to S$ be a closure operator on $S$.
Let $x \in S$.
By the definition of closure operator, $\cl$ is idempotent.
Thus by Fixed Point of Idempotent Mapping:
- An element of $S$ is a fixed point of $\cl$ if and only if it is in the image of $\cl$.
Thus the above definitions are equivalent.
$\blacksquare$