An introduction to Cp-theory and Grothendieck spaces
This is joint work with Prof. Frank Tall. An old theorem of Grothendieck states that countably compact subspaces of Cp(X) have compact closure whenever X is countably compact. We shall survey some basic results in Cp-theory and then focus on Grothendieck spaces, i.e. those for which the conclusion of Grothendieck theorem holds. Following Arhangel'skiĭ, we shall introduce the Lindelöf Σ-spaces, which belong to a wide class of well-behaved spaces, and prove that they are Grothendieck. We will show a compactness criterion for Fréchet-Urysohn Grothendieck spaces involving exchanging (ultra)limits which is similar to the classical Ptak's lemma. In a later occasion, we shall show the relevance of this results in Model Theory, e.g. the definability of pathological Banach spaces in various continuous logics.