Atoms in the lattice of covering operators in compact Hausdorff spaces

摘要

Let Comp be the category of compact Hausdorff spaces with continuous maps. A cover of a space in Comp is an irreducible preimage; equivalent covers are identified. A covering operator (co) is a function c assigning to each X in Comp a cover X -(cx) cX which is minimum among covers Y of X with Y = cY. The family of all such c is denoted coComp. This is a complete lattice (albeit a proper class), with bottom the identity operator id and top the Gleason (extremally disconnected, projective) cover operator g. Here, we completely determine the atoms (minimal elements above id) in the lattice coComp, show that any c not equal id in coComp is above an atom, and show that coComp is not atomic. At the end, we make some remarks about what the present paper does and does not tell us about several other categories related to Comp. (C) 2020 Elsevier B.V. All rights reserved.