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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.
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