The Semi-Baire-like Three-Space Problems

Abstract

Locally convex Baire spaces have poor permanence properties, as demonstrated by Valdivia's non-Baire product of two Baire spaces, which also proves that Baire is not a three-space property. The opposite is true of Baire-like (BL) and others in a long list of weaker space properties whose three-space status is known. We add semi-Baire-like (sBL) to the list. It has long been known that quasi-Baire (QB) is a three-space property, since a space is QB if and only if it is barrelled and non-Sσ, and the latter two are three-space properties. Since BL spaces are just those barrelled spaces that are sBL, the result would be similarly immediate for BL if sBL were a three-space property. Such a proof is impossible, for we show that sBL is not a three-space property, and much more: in the extended context, our results answer with optimal efficiency all remaining 631 questions of the form: “if F has property a and E/F has property b, then must the third space E have property c?”