S and L Spaces
Sara Leanne Mastros · D-Scholarship@Pitt (University of Pittsburgh) · 2009
An S-space is any topological space which is hereditarily separable but not Lindelof. An L-space, on the other hand, is hereditarily Lindelof but not separable. For almost a century, determining the necessary and suffcient conditions for the existence of these two kinds of spaces has been a fruitful area of research at the boundary of topology and axiomatic set theory. For most of that time, the twoproblems were imagined to be dual; that is, it was believed that the same setsof conditions that required or precluded one type would suffice for the other aswell. This, however, is not the case. When Todorcevic proved in 1981 that itis consistent, under ZFC, for no S-spaces to exist, everyone expected a similarresult to follow for L-spaces as well. Justin Tatch Moore surprised everyonewhen, in 2005, he constructed an L-space in ZFC. This paper summarizes andcontextualizes that result, along with several others in the field.