Normality of product spaces. II.

*(English)*Zbl 0699.54004
Topics in general topology, North-Holland Math. Libr. 41, 121-160 (1989).

[For the entire collection see Zbl 0684.00017.]

This is the second part of a survey of normality in products. The first part is by M. Atsuji (see the preceding review). This paper goes into detail about Morita’s P-spaces by proving Morita’s result of 1964 that \(X\times Y\) is normal for every metric space Y with weight w(Y)\(\leq {\mathfrak m}\) (where \({\mathfrak m}\geq \aleph_ 0)\), if and only if X is a normal P(\({\mathfrak m})\)-space, and related results. A number of mapping theorems are given. For example, the result of M. E. Rudin and M. Starbird [General Topol. Appl. 5, 235-248 (1975; Zbl 0305.54010)] that if f: \(X\to Y\) is a closed map, and Z is compact such that \(X\times Z\) is normal then \(Y\times Z\) is normal, and the analogous result of A. Bešlagić [Topology Appl. 22, 71-82 (1986; Zbl 0578.54017)] that if f: \(X\to Y\) is a closed map, and Z is paracompact M-space such that \(X\times Z\) is collectionwise normal then \(Y\times Z\) is collectionwise normal. The paper also gives most of what is known about product of normal spaces and Lashnev spaces \((=\) closed continuous image of a metric space). For example, T. Hoshina [Fundam. Math. 124, 143-153 (1984; Zbl 0567.54006)] proved that if X is normal and countably paracompact, and Y Lashnev, then \(X\times Y\) is normal iff it is countably paracompact. Numerous other results are also given. \(\{\) Taken together, the three survey papers (of Przymusinski, Atsuji and Hoshina) constitute a thorough survey of normality of products up to about 1988\(\}\).

This is the second part of a survey of normality in products. The first part is by M. Atsuji (see the preceding review). This paper goes into detail about Morita’s P-spaces by proving Morita’s result of 1964 that \(X\times Y\) is normal for every metric space Y with weight w(Y)\(\leq {\mathfrak m}\) (where \({\mathfrak m}\geq \aleph_ 0)\), if and only if X is a normal P(\({\mathfrak m})\)-space, and related results. A number of mapping theorems are given. For example, the result of M. E. Rudin and M. Starbird [General Topol. Appl. 5, 235-248 (1975; Zbl 0305.54010)] that if f: \(X\to Y\) is a closed map, and Z is compact such that \(X\times Z\) is normal then \(Y\times Z\) is normal, and the analogous result of A. Bešlagić [Topology Appl. 22, 71-82 (1986; Zbl 0578.54017)] that if f: \(X\to Y\) is a closed map, and Z is paracompact M-space such that \(X\times Z\) is collectionwise normal then \(Y\times Z\) is collectionwise normal. The paper also gives most of what is known about product of normal spaces and Lashnev spaces \((=\) closed continuous image of a metric space). For example, T. Hoshina [Fundam. Math. 124, 143-153 (1984; Zbl 0567.54006)] proved that if X is normal and countably paracompact, and Y Lashnev, then \(X\times Y\) is normal iff it is countably paracompact. Numerous other results are also given. \(\{\) Taken together, the three survey papers (of Przymusinski, Atsuji and Hoshina) constitute a thorough survey of normality of products up to about 1988\(\}\).

Reviewer: J.E.Vaughan

##### MSC:

54B10 | Product spaces in general topology |

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

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

54E20 | Stratifiable spaces, cosmic spaces, etc. |