The Lattice of Compactifications of a Locally Compact Space

Abstract

This is an expanded version of [5] by Magill. The results of [5] are proven with greater detail and any result stated in [5] but not proven is proven here. Let K (X) and K (Y) be used to indicate the lattice of Hausdorff compactifications of locally compact, non-compact spaces X and Y with X and Y Tychonoff. This paper primarily concerns how a lattice isomorphism between K (X) and K (Y) exists if and only if a homeomorphism between particular extensions of X and Y exists with specified properties. On the way to proving the main results, we prove several lemmas about β − families of compact extensions of Tychonoff spaces. Some of the Lemmas slightly generalize corresponding lemmas in [5]. Efforts are made to make this paper self- contained.