Interior 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 interior is an interior operator: the interior lies in $A$, grows with $A$, and is open, so taking it twice changes nothing. On $\R$, the map $i(x) = x - 1$ is deflationary and monotone but not idempotent, since $i(i(0)) = -2 \ne -1 = i(0)$. A map $i$ is an interior operator on $P$ if and only if it is a closure operator on the dual of $P$. On a power set, $A \mapsto X \setminus c(X \setminus A)$ is an interior operator for each closure operator $c$. Other names are kernel operator and dual closure operator.