The ideal structure of \(X^X\) (Q788842)

Given a Hausdorff topological space \(X\), the set \(X^X\) of functions from \(X\) to \(X\) with the product topology is a right topological semigroup under composition. The spaces \(X^X\) form probably the simplest class of right topological semigroups that are not semitopological. The spaces \(X^X\) also arise naturally in topological dynamics. In this paper we study the ideal structure of \(X^X\) in terms of certain closure operators on \(\Pi(X)\), the set of partitions of \(X\), and on \(P_+(X)\), the set of non-empty subsets of \(X\). We succeed in describing the right ideals, closed right ideals, minimal right ideals, left ideals, closed left ideals, and the (unique) minimal left ideal of \(X^X\).