Partition problems in topology.

*(English)*Zbl 0659.54001
Contemporary Mathematics, 84. Providence, RI: American Mathematical Society (AMS). xi, 116 p. (1989).

An S space is regular, hereditarily separable, non-Lindelöf; an L space is regular, hereditarily Lindelöf non-separable. The S and L space problems are: do S or L spaces exist?

The S and L space problems were, in the 1970’s, among the most active and frustrating problems in set-theoretic topology. Dating back, in some sense, to Suslin’s hypothesis (a Suslin line is, in particular, an L space) once sufficient set theory was in place a small industry was created in which CH, \(\diamond\), Suslin trees, Luzin sets and so on were used to construct S and L spaces not just for their own sake, but to answer questions in other areas of topology ranging from measure theory to manifolds. But do they exist without special hypotheses? Kunen and Szentmiklossy showed that it was possible to rule out particular S and L spaces under, say, Martin’s Axiom, but in general the answer was not known.

One reason mathematicians were attracted to these problems was the sense of strong combinatorial connections; S and L seemed to be barely disguised set theory, but no one could penetrate the disguise. Finally, in the early ’80’s, Todorčević did, showing the consistency of ZFC \(+\) there are no S spaces.

This book is the distillation of his ruminations connected with S and L. He describes it as dealing with “a topological version of the Ramsey theorem for the uncountable.” The reader should be assured that things are not quite so simple; the particular topological version of Ramsey’s theorem referred to appears on p. 71 of 101 pages of text and is by no means the only combinatorial principle considered.

Todorčević’s career has been characterized by a deep vision of combinatorics on small uncountable cardinals, which he refines and returns to, proving theorems in other areas of mathematics with elegance and grace. He is, for example, the master of the Aronszajn tree (of which the reader sees a small glimpse in this book) as well as one of the masters of Ramsey theory in its broadest sense. Before getting to the point of the book (the consistency of “there are L spaces but no S spaces”) he takes the reader through a wide range of results including (i) technical work on oscillating real numbers (how many times do two functions in \(\omega^{\omega}\) cross over each other? This refines the analysis of functions under the relation of eventual dominance); (ii) ZFC theorems (choosing one at random to give the flavor: there is a compact 0-dimensional space whose cellularity is less than the cellularity of its square); (iii) theorems involving simple set theoretic hypotheses on the reals; (iv) theorems needing more difficult set theoretic hypotheses, given here as PFA results but, in fact, not needing large cardinals for their proof.

In keeping with its being driven by the use of set theory, rather than by topological considerations, the results proved in this book involve concepts ranging over the gamut of topology, from cardinal invariants to perfect normality cometrizability. They include many theorems by other mathematicians, often with new proofs, and there are complete and seemingly impeccable historial notes at the end of each chapter. The proofs sometimes are written in a kind of shorthand, leaving many details to the reader, but the outline is always clear, and the exercise is good for us. This monograph is applied set theory at its best. The ambitious reader might note that “there is no L space” is still not known to be consistent.

The S and L space problems were, in the 1970’s, among the most active and frustrating problems in set-theoretic topology. Dating back, in some sense, to Suslin’s hypothesis (a Suslin line is, in particular, an L space) once sufficient set theory was in place a small industry was created in which CH, \(\diamond\), Suslin trees, Luzin sets and so on were used to construct S and L spaces not just for their own sake, but to answer questions in other areas of topology ranging from measure theory to manifolds. But do they exist without special hypotheses? Kunen and Szentmiklossy showed that it was possible to rule out particular S and L spaces under, say, Martin’s Axiom, but in general the answer was not known.

One reason mathematicians were attracted to these problems was the sense of strong combinatorial connections; S and L seemed to be barely disguised set theory, but no one could penetrate the disguise. Finally, in the early ’80’s, Todorčević did, showing the consistency of ZFC \(+\) there are no S spaces.

This book is the distillation of his ruminations connected with S and L. He describes it as dealing with “a topological version of the Ramsey theorem for the uncountable.” The reader should be assured that things are not quite so simple; the particular topological version of Ramsey’s theorem referred to appears on p. 71 of 101 pages of text and is by no means the only combinatorial principle considered.

Todorčević’s career has been characterized by a deep vision of combinatorics on small uncountable cardinals, which he refines and returns to, proving theorems in other areas of mathematics with elegance and grace. He is, for example, the master of the Aronszajn tree (of which the reader sees a small glimpse in this book) as well as one of the masters of Ramsey theory in its broadest sense. Before getting to the point of the book (the consistency of “there are L spaces but no S spaces”) he takes the reader through a wide range of results including (i) technical work on oscillating real numbers (how many times do two functions in \(\omega^{\omega}\) cross over each other? This refines the analysis of functions under the relation of eventual dominance); (ii) ZFC theorems (choosing one at random to give the flavor: there is a compact 0-dimensional space whose cellularity is less than the cellularity of its square); (iii) theorems involving simple set theoretic hypotheses on the reals; (iv) theorems needing more difficult set theoretic hypotheses, given here as PFA results but, in fact, not needing large cardinals for their proof.

In keeping with its being driven by the use of set theory, rather than by topological considerations, the results proved in this book involve concepts ranging over the gamut of topology, from cardinal invariants to perfect normality cometrizability. They include many theorems by other mathematicians, often with new proofs, and there are complete and seemingly impeccable historial notes at the end of each chapter. The proofs sometimes are written in a kind of shorthand, leaving many details to the reader, but the outline is always clear, and the exercise is good for us. This monograph is applied set theory at its best. The ambitious reader might note that “there is no L space” is still not known to be consistent.

Reviewer: J.Roitman

##### MSC:

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

54A35 | Consistency and independence results in general topology |