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

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
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