Representation Theorems for Connected Compact Hausdorff Spaces - Džamonja, Mirna (Sarajevo Journal of Mathematics, 2024)
We present two theorems which can be used to represent compact connected Hausdorff spaces in an algebraic context, using a Stone-like representation. The first theorem stems from the work of Wallman and shows that every distributive disjunctive normal lattice is the lattice of closed sets in a unique up to homeomorphism connected compact Hausdorff space. The second theorem stems from the work of Jung and Sünderhauf. Introducing the notion of strong proximity involution lattices, it shows that every such lattice can be uniquely represented as the lattice of pairs of compact and open sets of connected compact Hausdorff space. As a consequence we easily obtain a somewhat surprising theorem birepresenting distributive disjunctive normal lattices and strong proximity involution lattices. 2000 Mathematics Subject Classification. Primary: 06D05, 54H10; Secondary: 68R99