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**Cardinal functions. I.**
*(English)*
Zbl 0559.54003

Handbook of set-theoretic topology, 1-61 (1984).

[For the entire collection see Zbl 0546.00022.]

During the 1920’s Alexandroff und Urysohn proved that every compact perfectly normal space has cardinality at most continuum. This result motivated the question whether every first countable compact space has cardinality at most continuum. In 1969, Arkhangelskij answered this question in the affirmative. In fact, he showed that every first countable Lindelöf space has cardinality at most continuum and, more generally, that the cardinality of any Hausdorff space never exceeds exp(L(X).\(\chi\) (X)), where L(X) and \(\chi\) (X) denote the Lindelöf degree and the character of X, respectively. The formulation of Arkhangelskij’s result in the language of cardinal functions is quite important. It focusses attention on two natural cardinal functions associated to any space X, namely L(X) and \(\chi\) (X), and it clearly illustrates that these cardinal functions determine an upper bound for another cardinal function on X, namely its cardinality. One naturally wonders whether there are other relations between naturally arising cardinal functions on topological spaces. It turns out that there are many relations many of which are important. The author carefully defines the most important cardinal functions and states the relationships between these functions. In addition, he treats in detail cardinal functions on the class of all compact spaces and on the class of all metrizable spaces. He concentrates on results that can be proved (or disproved) in ZFC. This is a valuable survey paper.

During the 1920’s Alexandroff und Urysohn proved that every compact perfectly normal space has cardinality at most continuum. This result motivated the question whether every first countable compact space has cardinality at most continuum. In 1969, Arkhangelskij answered this question in the affirmative. In fact, he showed that every first countable Lindelöf space has cardinality at most continuum and, more generally, that the cardinality of any Hausdorff space never exceeds exp(L(X).\(\chi\) (X)), where L(X) and \(\chi\) (X) denote the Lindelöf degree and the character of X, respectively. The formulation of Arkhangelskij’s result in the language of cardinal functions is quite important. It focusses attention on two natural cardinal functions associated to any space X, namely L(X) and \(\chi\) (X), and it clearly illustrates that these cardinal functions determine an upper bound for another cardinal function on X, namely its cardinality. One naturally wonders whether there are other relations between naturally arising cardinal functions on topological spaces. It turns out that there are many relations many of which are important. The author carefully defines the most important cardinal functions and states the relationships between these functions. In addition, he treats in detail cardinal functions on the class of all compact spaces and on the class of all metrizable spaces. He concentrates on results that can be proved (or disproved) in ZFC. This is a valuable survey paper.

Reviewer: J.van Mill

### MSC:

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

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

54D30 | Compactness |

54E35 | Metric spaces, metrizability |