An introduction to \(\beta\omega\).

*(English)*Zbl 0555.54004
Handbook of set-theoretic topology, 503-567 (1984).

[For the entire collection see Zbl 0546.00022.]

The Stone-Čech compactification of the integers is denoted by \(\beta\omega\) whereas the remainder \((=\beta\omega-\omega)\) is denoted by \(\omega^*\). The paper under review is a very good guide to the theory of \(\beta\omega\) and \(\omega^*\). The main topics studied here are the following: characterizations of \(\omega^*\), continuous images and closed subspaces of \(\omega^*\), automorphisms and \(C^*\)-embedded subspaces of \(\omega^*\), retracts of \(\beta\omega\) and \(\omega^*\), P- points and nowhere dense P-sets in \(\omega^*\). These aspects of \(\beta\omega\) and \(\omega^*\) are studied under CH as well as under nonCH plus some additional set-theoretic assumptions (e.g. Martin’s axiom. Some of the results are proved in ZFC. In particular, Kunen’s theorem that there exist weak P-points in \(\omega^*\) is presented. The paper also contains a list of the most interesting open problems. One of them was recently answered by Petr Simon, who proved (in ZFC) that there exists a separable closed subset of \(\omega^*\) which is not a retract of \(\omega^*\). The paper well is written and contain a fair amount of recent results. Since there is still much to do in this area of topology, I strongly recommend this paper to everybody interested in set-theoretic topology.

The Stone-Čech compactification of the integers is denoted by \(\beta\omega\) whereas the remainder \((=\beta\omega-\omega)\) is denoted by \(\omega^*\). The paper under review is a very good guide to the theory of \(\beta\omega\) and \(\omega^*\). The main topics studied here are the following: characterizations of \(\omega^*\), continuous images and closed subspaces of \(\omega^*\), automorphisms and \(C^*\)-embedded subspaces of \(\omega^*\), retracts of \(\beta\omega\) and \(\omega^*\), P- points and nowhere dense P-sets in \(\omega^*\). These aspects of \(\beta\omega\) and \(\omega^*\) are studied under CH as well as under nonCH plus some additional set-theoretic assumptions (e.g. Martin’s axiom. Some of the results are proved in ZFC. In particular, Kunen’s theorem that there exist weak P-points in \(\omega^*\) is presented. The paper also contains a list of the most interesting open problems. One of them was recently answered by Petr Simon, who proved (in ZFC) that there exists a separable closed subset of \(\omega^*\) which is not a retract of \(\omega^*\). The paper well is written and contain a fair amount of recent results. Since there is still much to do in this area of topology, I strongly recommend this paper to everybody interested in set-theoretic topology.

Reviewer: A.Błaszczyk

##### MSC:

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

54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |

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

54A35 | Consistency and independence results in general topology |

54D40 | Remainders in general topology |