Distinguished Cp (X) spaces and the strongest locally convex topology

Abstract

Since Tychonoff spaces X serve as continuous Hamel bases for the strong dual Lβ (X) of Cp (X), an old splitting theorem proves: Cp (X) is distinguished ⇔ Lβ (X) has the strongest locally convex topology (slctop) [Ferrando/Ka˛kol]. Our new splitting theorem: The span LX (Y ) of Y ⊆ X complements LX (X\Y ) in Lβ (X). Thereby we prove If X = Y1 ∪· · ·∪Yn and each Cp Yj is distinguished, then so is Cp (X), provided either (i) all Yj are Gδ sets, or (ii) all are Fσ sets. Hence, provided (iii) all Yj are open, or (iv) all are closed. Parts (ii)/(iv) extend to countable unions (known). Part (i) does not, via Michael’s line. Countable case (iii) remains open. A dozen recent related results are proved/improved in our slctop analysis of Lβ (X).