A note on the lattices $DP(X)$ and $K(X)$

Abstract

Using the order structure of the lattice $DP(X)$ of density preserving continuous maps on a Hausdorff space $X$ without isolated points, we describe closed nowhere dense subsets of $X$ and, for a subspace $A$ of $X$, we also deduce topological properties of the space $X-A$ from the lattice theoretic properties of $DP(X,A)$. Finally, we use them to obtain Thrivikraman's results concerning $\beta X-X$ and $K(X)$ and, Magill's result concerning the automorphism group of the lattice $K(X)$.