Set-theoretic Problems Concerning Lindelöf Spaces

I survey problems concerning Lindelöf spaces which have partial settheoretic solutions. Lindelöf spaces, i.e. spaces in which every open cover has a countable subcover, are a familiar class of topological spaces. There is a significant number of (mainly classic) problems concerning Lindelöf spaces which are unsolved, but have partial set-theoretic solutions. For example, consistency is known but independence is not; large cardinals suffice but are not known to be necessary, and so forth. The purpose of this note — which is an expanded version of a talk given at the 2010 BEST conference — is to survey such questions in the hope that set theorists will find them worthy of attention. Indeed we strongly suspect that the difficulty of these problems is more set-theoretic than topological. Not much topological knowledge is needed to work on them. Undefined terms can be found in [Eng89]. In Sections 1–3 we shall assume all spaces are T3, except for a remark at the end of Section 2. I shall start with a collection of problems I have been investigating for the past couple of years. Several of these problems are classic and well-known; other are more specialized or recent, but are related to the classic ones. 1 Productively Lindelöf spaces Definition 1.1. A space X is productively Lindelöf if X × Y is Lindelöf, for every Lindelöf Y . X is powerfully Lindelöf if X is Lindelöf. Both of these concepts have been studied for a long time, but the terminology is recent: [BKR07], [AT] respectively. Ernie Michael wondered more than thirty years ago whether: 1The author was supported in part by NSERC Grant A-7354. (2010) Mathematics Subject Classification. Primary 54A35, 54D20; Secondary 54A25, 54D30.