Characteristics of Complexity within the Lattice of Compactifications

Abstract

The purpose of this research is to determine some topological characteristics that may be used to classify a Hausdorff compactification of a topological space as a complex compactification, within the lattice of compactifications. The Stone-Cech compactification is the supremum of the lattice, the Alexandroff one-point compactification the infimum. We look to characteristics that the Stone-Cech compactification holds and whether or not those properties are found in compactifications "close" to it. The idea of a complex compactification has not been strictly defined and there are numerous properties that could be used in a definition.

Beginning with mappings with a finite number of nontrivial fibers,, we find that F-space is invariant. F-space can not be guaranteed for all finite-to-one mappings. The characteristic we call G-int under any finite-to-one irreducible mapping and the continuous image of a nowhere F space is nowhere F, a characteristic of compactifications that are simple. We also consider the mappings on the Stone-Cech compactification of the natural numbers that are simple mappings, proving that if a simple mapping is finite-to-one, then so is its generator and vice versa.