Normality versus collectionwise normality.

*(English)*Zbl 0552.54011
Handbook of set-theoretic topology, 685-732 (1984).

[For the entire collection see Zbl 0546.00022.]

Collectionwise normality is, in general, stronger than normality since it says that every discrete collection of pairwise disjoint closed sets can be separated regardless of the cardinality of the collection while normality says that every two disjoint closed sets can be separated. (A collection of pairwise disjoint closed sets is discrete if the union of each subcollection is closed.) This chapter in the ”Handbook” is mainly concerned with the question: Under what conditions does normality imply collectionwise normality? The answer may involve topological conditions or both topological and set-theoretic conditions.

The author divides the paper into four sections: I. Theorems, II. Examples, III. Problems, and IV. Historical Notes. Proofs are given for most of the theorems, especially those which have not already appeared in print. A rather complete bibliography is included.

Section I is divided into four parts. The first part consists of theorems which require no special set-theoretic axioms or techniques, while the second and third parts do require additional set-theoretic axioms and techniques. The second part is ”essentially concerned with points while the third deals with arbitrary closed sets.” In the fourth part of Section I, the author applies the preceding theorems to the normal Moore space metrization problem: when is a normal Moore space metrizable and when not ? Since every collectionwise normal Moore space is metrizable (Bing) one can clearly see the relevance of the theorems in this paper to this problem the solution of which (according to the author) ”has stimulated - rather than killed - activity in this area.” Coupled with W. G. Fleissner’s chapter, ”The normal Moore space conjecture and large cardinals”, ibid., 733-760 (1984), one has a rather complete picture of the area, ”an area which has for fifteen years been at the cutting edge of set-theoretic topology.”

Collectionwise normality is, in general, stronger than normality since it says that every discrete collection of pairwise disjoint closed sets can be separated regardless of the cardinality of the collection while normality says that every two disjoint closed sets can be separated. (A collection of pairwise disjoint closed sets is discrete if the union of each subcollection is closed.) This chapter in the ”Handbook” is mainly concerned with the question: Under what conditions does normality imply collectionwise normality? The answer may involve topological conditions or both topological and set-theoretic conditions.

The author divides the paper into four sections: I. Theorems, II. Examples, III. Problems, and IV. Historical Notes. Proofs are given for most of the theorems, especially those which have not already appeared in print. A rather complete bibliography is included.

Section I is divided into four parts. The first part consists of theorems which require no special set-theoretic axioms or techniques, while the second and third parts do require additional set-theoretic axioms and techniques. The second part is ”essentially concerned with points while the third deals with arbitrary closed sets.” In the fourth part of Section I, the author applies the preceding theorems to the normal Moore space metrization problem: when is a normal Moore space metrizable and when not ? Since every collectionwise normal Moore space is metrizable (Bing) one can clearly see the relevance of the theorems in this paper to this problem the solution of which (according to the author) ”has stimulated - rather than killed - activity in this area.” Coupled with W. G. Fleissner’s chapter, ”The normal Moore space conjecture and large cardinals”, ibid., 733-760 (1984), one has a rather complete picture of the area, ”an area which has for fifteen years been at the cutting edge of set-theoretic topology.”

Reviewer: F.B.Jones

##### MSC:

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

54E30 | Moore spaces |

54E35 | Metric spaces, metrizability |