INTRODUCTION In this memoir we present results closely related to the well-known con- jecture of general topology asserting that the following two properties are equivalent for every regular space X: (HS) Every subset of X contains a countable subset with the same closure. (HL) Every family of open subsets of X contains a countable subfamily with the same union. The statement that (HS) implies (HL) will be denoted by (S) and the statement that (HL) implies (HS) will be denoted by (L). In the vast literature on this subject these two statements are usually called the "S-space problem" and the "L-space problem," respectively. The separate consideration of the two implications may seem unnatural at first, but one of the results of this memoir is that (S) and (L) are actually quite different problems and that the methods designed for solving one of them do not work for the other. The statements (S) and (L) are claiming the equivalence of two quite di- verse topological properties and a priori one sees no reason why they should be true. But the problems seem to be very basic for general topology be- cause they are connected to many subjects of this field (see [21], [37], [41]). For example, one can hardly find an expert in this field not having a result touching (S) and (L). Quite often when working on seemingly unrelated prob- lems in their field they encounter a difficulty which can be resolved if one knows the answer to certain instances of (S) and (L). Even more often, the proof technique designed for solving (S) and (L) turns out to be useful in many other problems of general topology. Why is this so? Well, this is so because Ramsey-type theorems are basic and so much needed in many parts of mathematics and (S) and (L) happen to be Ramsey-type properties of the uncountable most often needed by the general topologist. To understand this point one only needs to remember that (S) and (L) have simple translations (see §7) asserting the existence of uncountable homogeneous sets for certain partitions of [wJ]2 into two sets. This also explains the reason why these two topological questions demand so much from set theory for their solution. In http://dx.doi.org/10.1090/conm/084
Previous Page Next Page