Lúcia R. Junqueira

The topology of elementary submodels

Abstract Given a topological space 〈 X, T 〉 , we take an elementary submodel M of a sufficiently large initial fragment of the universe containing 〈 X, T 〉 and naturally define a space XM. We consider under what circumstances XM is a (nice image of a) subspace of X, and what properties of X are retained by XM.

On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω1-monolithic compact space X, if Cp(X)is star countable then it is Lindelöf.

Forcing and normality

Abstract We solve or partially solve a number of problems of Watson concerning the preservation of normality by forcing. We also provide general machinery for applying reflection methods in the class of spaces of pointwise countable type.

Compact spaces, elementary submodels, and the countable chain condition

Given a space 〈X,J〉 in an elementary submodel M of H(θ), define XM to be X∩M with the topology generated by {U∩M:U∈J∩M}. It is established, using anti-large-cardinals assumptions, that if XM is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X=XM. Assuming CH + SCH (the Singular Cardinals Hypothesis) in addition, the result holds for any compact XM satisfying the countable chain condition.

On Extensions of Elementary Submodels by Forcing

We study when a partial order ℙ preserves certain properties of an elementary submodel (of a structure of the form H θ ), including countable closure and ω-covering. In particular, we discuss the existence of ω-covering elementary submodels of different sizes.