Note about lindelof Sigma-SPACES nu X

Abstract

The paper deals with the following problem: characterize Tichonov spaces X whose realcompactification ¿X is a Lindelöf ¿-space. There are many situations (both in topology and functional analysis) where Lindelöf ¿ (even K-analytic) spaces ¿X appear. For example, if E is a locally convex space in the class fraktur G sign in sense of Cascales and Orihuela (fraktur G sign includes among others (LM)-spaces and (DF)-spaces), then ¿(E¿,¿(E¿,E)) is K-analytic and E is web-bounded. This provides a general fact (due to Cascales-Kakol-Saxon): if E¿fraktur G sign, then ¿(E¿,E) is K-analytic if and only if ¿(E¿,E) is Lindelöf. We prove a corresponding result for spaces Cp(X) of continuous real-valued maps on X endowed with the pointwise topology: ¿X is a Lindelöf ¿-space if and only if X is strongly web-bounding if and only if Cp(X) is web-bounded. Hence the weak* dual of C p(X) is a Lindelöf ¿-space if and only if Cp(X) is web-bounded and has countable tightness. Applications are provided. For example, every E¿fraktur G sign is covered by a family {A¿ :¿¿¿} of bounded sets for some nonempty set ¿¿¿¿. © Copyright Australian Mathematical Publishing Association Inc. 2011.