An algebraic and logical approach to continuous images. (English) Zbl 1169.54354

Hart, Klaas Pieter

Acta Univ. Carol., Math. Phys. 43, No. 2, 5-25 (2002).

Summary: Continuous mappings between compact Hausdorff spaces can be studied using homomorphisms between algebraic structures (lattices, Boolean algebras) associated with the spaces. This gives us more tools with which to tackle problems about these continuous mappings – also tools from Model Theory. We illustrate by showing that 1) the Čech-Stone remainder \([0,\infty)\) has a universality property akin to that of \(\mathbb N^*\); 2) a theorem of Maćkowiak and Tymchatyn implies it own generalization to non-metric continua; and 3) certain concrete compact spaces need not be continuous images of \(\mathbb N^*\).

MSC:

54F15	Continua and generalizations
03C50	Models with special properties (saturated, rigid, etc.)
03C98	Applications of model theory
03E55	Large cardinals
03E65	Other set-theoretic hypotheses and axioms
06D05	Structure and representation theory of distributive lattices
06E15	Stone spaces (Boolean spaces) and related structures
28A60	Measures on Boolean rings, measure algebras
54A35	Consistency and independence results in general topology
54C10	Special maps on topological spaces (open, closed, perfect, etc.)
54D40	Remainders in general topology
54D80	Special constructions of topological spaces (spaces of ultrafilters, etc.)
54F45	Dimension theory in general topology
54H10	Topological representations of algebraic systems

Keywords:

lattice; ultrafilter; Wallman representation; Wallman space; Boolean algebra; Stone space; duality; continuous surjection; embedding; universal continuum; Čech-Stone remainder of \(\mathbb R\)