Compact spaces and their generalizations.

*(English)*Zbl 0801.54001
Hušek, Miroslav (ed.) et al., Recent progress in general topology. Papers from the Prague Toposym 1991, held in Prague, Czechoslovakia, Aug. 19-23, 1991. Amsterdam: North-Holland. 571-640 (1992).

This survey paper presents results concerning compact spaces and their generalizations which have been obtained in the decade prior to 1992. The discussion is broken into topics: The Moore-Mrówka problem; In between countable tightness and first-countability: the fine structure; Points of small character in countably tight and sequential compact spaces; Comparing properties of \(X\) with that of \(X^ 2\) and \(X^ 2\setminus \Delta\); Topological problems arising from functional analysis; Dungundji spaces; dyadic spaces, \(\kappa\)-metrizable spaces and their generalizations; Dense subspaces of compact spaces; Generalizations of compact spaces; and Miscellanea.

Some of these topics focus on dominant questions that were still open a decade ago (such as Moore-Mrówka) and present solutions, partial solutions, related theorems and new open questions. Other topics (such as topological problems arising from functional analysis) are less focused and look at work done on a variety of new and old questions in the area. For topics which already have detailed surveys (such as Eberlein, Gul’ko, and Corson compact space or the Scarborough-Stone problem) the reader is referred to the survey with little discussion. The organization of the paper puts the work of the 1980’s and early 1990’s into a context that would be beneficial to a newcomer in the field. It also introduces western readers to the work of Soviet topologists, through both the exposition and the extensive bibliography.

In addition to reciting the recent progress on compact spaces, this article traces the historical development of several problems. Most notable are the discussions of the Moore-Mrówka problem and Katětov’s 1948 question of whether a compact space \(X\) is metrizable if the square \(X^ 2\) is hereditarily normal.

For the entire collection see [Zbl 0782.00072].

Some of these topics focus on dominant questions that were still open a decade ago (such as Moore-Mrówka) and present solutions, partial solutions, related theorems and new open questions. Other topics (such as topological problems arising from functional analysis) are less focused and look at work done on a variety of new and old questions in the area. For topics which already have detailed surveys (such as Eberlein, Gul’ko, and Corson compact space or the Scarborough-Stone problem) the reader is referred to the survey with little discussion. The organization of the paper puts the work of the 1980’s and early 1990’s into a context that would be beneficial to a newcomer in the field. It also introduces western readers to the work of Soviet topologists, through both the exposition and the extensive bibliography.

In addition to reciting the recent progress on compact spaces, this article traces the historical development of several problems. Most notable are the discussions of the Moore-Mrówka problem and Katětov’s 1948 question of whether a compact space \(X\) is metrizable if the square \(X^ 2\) is hereditarily normal.

For the entire collection see [Zbl 0782.00072].

Reviewer: D.Mooney (Bowling Green)

##### MSC:

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

54D30 | Compactness |

54D55 | Sequential spaces |

54A35 | Consistency and independence results in general topology |

54D60 | Realcompactness and realcompactification |

46A11 | Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) |

54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |

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

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