A Boolean algebra and a Banach space obtained by push-out iteration

Under the assumption that the continuum c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(N)/fin has under CH and in the aleph2-Cohen model. We prove a similar result in the category of Banach spaces

An Algebraic and Logical approach to continuous images

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 the \v{C}ech-Stone remainder $[0,\infty)$ has a universality property akin to that of $N^*$; a theorem of Ma\'ckowiak and Tymchatyn implies it own generalization to non-metric continua; and certain concrete compact spaces need not be continuous images of $N^*$.Comment: Notes from a series of lectures at http://www.cts.cuni.cz/events/ws/2002/ws2002.htm, the 30th Winter School on Abstract Analysis 2002-05-02: corrected version after referee's repor

Reduced Coproducts of Compact Hausdorff Spaces

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the reduced coproduct , which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the ultracoproduct can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces