Closure operator

Remarks

Let $X$ be a topological space and $P$ the set of subsets of $X$ ordered by inclusion. The map sending $A$ to its closure $\overline{A}$ is a closure operator: $A \subseteq \overline{A}$, $A \subseteq B$ gives $\overline{A} \subseteq \overline{B}$, and $\overline{\overline{A}} = \overline{A}$. On $\R$, the map $c(x) = x + 1$ is extensive and monotone but not idempotent, since $c(c(0)) = 2 \ne 1 = c(0)$. The three clauses together are equivalent to one condition: $x \le c(y)$ if and only if $c(x) \le c(y)$, for all $x, y \in P$. The closed elements determine $c$: $c(x)$ is the least closed element above $x$. If $(f, g)$ is a Galois connection with lower adjoint $f$, then $g \circ f$ is a closure operator. The dual notion is an interior operator.