Modern general topology. 2nd rev. ed.

*(English)*Zbl 0598.54001
North-Holland Mathematical Library, Vol. 33, Amsterdam-New York-Oxford: North-Holland. X, 522 p. $ 92.50; Dfl. 250.00 (1985).

This is a distinguished book by a distinguished mathematician. The range is selective rather than encyclopedic, reflecting the author’s many areas of interest and competence; these are impressive in their breadth, but (in this reviewer’s opinion) they fall short nevertheless of a complete dose of ”modern general topology”. For a sympathetic, careful, authoritative account of paracompactness and metrizability theory (to which of course the author himself has made fundamental contributions of lasting value) one may recommend this book above others to both the novice and the expert. Collectionwise normality, theorems of Tamano type involving (for example) normality of \(X\times Y\) for every compact space Y, uniform spaces, proximity spaces, the P-spaces of Morita, Frink’s Wallman problem (but without mention of relevant contributions of Banashewski, and without the detals of Ul’janov’s coup de grâce), the theory of selections, inverse limits and direct limits - these and many other honorable topics receive skillful treatment at the author’s hand. The major constructs which are the sine qua non of an introductory course in topology - Tychonoff’s product theorem, Urysohn’s lemma, Tietze’s theorem, the Baire category theorem, the Čech-Stone compactification, the Stone-Weierstrass theorem - are also available here.

Although Martin’s axiom is mentioned, and a couple of examples are given of questions in general topology not settled by the usual axioms of ZFC, there is no systematic effort here to survey the effect of Cohen’s forcing technique and its derivatives on general topology, nor to describe competing models which determine particular problems differently. Likewise the author chooses to ignore the advances and simplifications in general topology achieved in recent years through categorical methods; indeed, the word category appears just once (in a footnote). Chapter VIII, which ”was added to fit the great developments of general topology that occurred since 1968 when the initial edition of this book appeared”, contains four sections titled: Linearly ordered spaces; Cardinal functions; Dyadic spaces; and Measures and topological spaces. This reviewer appreciated in particular the readable account of R. Pol’s proofs of the theorems of Arkhangel’ski\uj and Hajnal- Juhász: every Hausdorff space X satisfies \(| X| \leq 2^{\chi (X)\cdot d(X)}\) and \(| X| \leq 2^{\chi (X)\cdot c(X)}.\)

An initial perusal of this book gave rise to only two concrete criticisms. (1) The author needs and proves only the case \({\mathfrak m}=\aleph_ 1\) of the Čech-Pospíšil theorem, and this he attributes to a 1958 paper of Mrówka; the theorem in full generality dates back to 1938, however, so the original authors should have been cited. (2) By way of proof that the continuous image of a realcompact space need not be realcompact (p. 184), the author invokes the existence of a continuous one-to-one mapping from the discrete real line onto the space of countable ordinals in its order topology; the point could have been made more safely and accurately by replacing ”the discrete real line” with ”a discrete space of cardinality \(\chi_ 1''\), or by dropping the adjective ”one-to-one”.

The book is both a literate treatment by a world-class topologist of significant portions of Modern General Topology, and a personal judgemental statement as to what does and does not deserve to be recorded for posterity. On each count it is a valued addition to the literature.

Although Martin’s axiom is mentioned, and a couple of examples are given of questions in general topology not settled by the usual axioms of ZFC, there is no systematic effort here to survey the effect of Cohen’s forcing technique and its derivatives on general topology, nor to describe competing models which determine particular problems differently. Likewise the author chooses to ignore the advances and simplifications in general topology achieved in recent years through categorical methods; indeed, the word category appears just once (in a footnote). Chapter VIII, which ”was added to fit the great developments of general topology that occurred since 1968 when the initial edition of this book appeared”, contains four sections titled: Linearly ordered spaces; Cardinal functions; Dyadic spaces; and Measures and topological spaces. This reviewer appreciated in particular the readable account of R. Pol’s proofs of the theorems of Arkhangel’ski\uj and Hajnal- Juhász: every Hausdorff space X satisfies \(| X| \leq 2^{\chi (X)\cdot d(X)}\) and \(| X| \leq 2^{\chi (X)\cdot c(X)}.\)

An initial perusal of this book gave rise to only two concrete criticisms. (1) The author needs and proves only the case \({\mathfrak m}=\aleph_ 1\) of the Čech-Pospíšil theorem, and this he attributes to a 1958 paper of Mrówka; the theorem in full generality dates back to 1938, however, so the original authors should have been cited. (2) By way of proof that the continuous image of a realcompact space need not be realcompact (p. 184), the author invokes the existence of a continuous one-to-one mapping from the discrete real line onto the space of countable ordinals in its order topology; the point could have been made more safely and accurately by replacing ”the discrete real line” with ”a discrete space of cardinality \(\chi_ 1''\), or by dropping the adjective ”one-to-one”.

The book is both a literate treatment by a world-class topologist of significant portions of Modern General Topology, and a personal judgemental statement as to what does and does not deserve to be recorded for posterity. On each count it is a valued addition to the literature.

Reviewer: W.W.Comfort

##### MSC:

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

54E35 | Metric spaces, metrizability |

54C65 | Selections in general topology |

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |

54D60 | Realcompactness and realcompactification |

54E15 | Uniform structures and generalizations |