Basic S and L.

*(English)*Zbl 0594.54001
Handbook of set-theoretic topology, 295-326 (1984).

[For the entire collection see Zbl 0546.00022.]

The ”Handbook of set-theoretic topology” is a truly remarkable enterprise, where active research mathematicians were asked to write an introduction to their special fields of interest for ”the friendly neighborhood topologist”. ”Handbook” is somewhat an understatement for a book with ”1273” as the last page number. It contains a combination of surveys with references to recent results.

Set theoretic considerations have a surprising tendency to appear suddenly in the context of otherwise standard problems in topology and therefore this book can only be highly recommended to every research topologist.

The problem of the existence of S and L spaces, which is the topic of this article, has its origin in the question whether there is a useful relation between limit point properties (e.g. separability) and covering properties. M. Souslin asked in 1920 [Fundam. Math. 1, 223 (1920)] whether a linearly ordered space must be separable whenever it satisfies the countable chain condition. An S-space is hereditarily separable but not Lindelöf. An L-space is hereditarily Lindelöf, but not separable. Since there are trivial examples when only \(T_ 2\) is assumed, one usually adds \(T_ 3\) to the definition. A Souslin line would be a nice example of a regular L-space. Using CH several S- and L-spaces were constructed. \(MA+\neg CH\) killed those constructions. And this is the entry point of the set-theoretic connection. A tremendous amount of research activity was put into these difficult problems and several consistency results were obtained. This survey (together with the article: ”Martin’s axiom and first-countable S- and L-spaces” by U. Abraham and S. Todorčević, which is also included in this handbook [pp. 327-346; Zbl 0565.54005)] forms an excellent reference source for this field. It covers the main results up to Szentmiklossy’s result, that MA\(+\neg CH\) is consistent with the existence of L-spaces and Todorcevic’s that it is consistent that ”there are no L-spaces”. An extensive reference list adds to the usefulness of this article. To enjoy these chapters of the ”Handbook” completely, the reader should have a certain background in axiomatic set theory. If you are looking for an easier and more topological introduction, Mary Ellen Rudin’s very readable ”S and L spaces” [Surveys in general topology, 431-444 (1980; Zbl 0457.54015)] might be a nice place to start.

The ”Handbook of set-theoretic topology” is a truly remarkable enterprise, where active research mathematicians were asked to write an introduction to their special fields of interest for ”the friendly neighborhood topologist”. ”Handbook” is somewhat an understatement for a book with ”1273” as the last page number. It contains a combination of surveys with references to recent results.

Set theoretic considerations have a surprising tendency to appear suddenly in the context of otherwise standard problems in topology and therefore this book can only be highly recommended to every research topologist.

The problem of the existence of S and L spaces, which is the topic of this article, has its origin in the question whether there is a useful relation between limit point properties (e.g. separability) and covering properties. M. Souslin asked in 1920 [Fundam. Math. 1, 223 (1920)] whether a linearly ordered space must be separable whenever it satisfies the countable chain condition. An S-space is hereditarily separable but not Lindelöf. An L-space is hereditarily Lindelöf, but not separable. Since there are trivial examples when only \(T_ 2\) is assumed, one usually adds \(T_ 3\) to the definition. A Souslin line would be a nice example of a regular L-space. Using CH several S- and L-spaces were constructed. \(MA+\neg CH\) killed those constructions. And this is the entry point of the set-theoretic connection. A tremendous amount of research activity was put into these difficult problems and several consistency results were obtained. This survey (together with the article: ”Martin’s axiom and first-countable S- and L-spaces” by U. Abraham and S. Todorčević, which is also included in this handbook [pp. 327-346; Zbl 0565.54005)] forms an excellent reference source for this field. It covers the main results up to Szentmiklossy’s result, that MA\(+\neg CH\) is consistent with the existence of L-spaces and Todorcevic’s that it is consistent that ”there are no L-spaces”. An extensive reference list adds to the usefulness of this article. To enjoy these chapters of the ”Handbook” completely, the reader should have a certain background in axiomatic set theory. If you are looking for an easier and more topological introduction, Mary Ellen Rudin’s very readable ”S and L spaces” [Surveys in general topology, 431-444 (1980; Zbl 0457.54015)] might be a nice place to start.

Reviewer: E.R.Unger

##### MSC:

54A35 | Consistency and independence results in general topology |

57R65 | Surgery and handlebodies |

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |

03E35 | Consistency and independence results |