##
**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 114.14102

**Autor: ** Erdös, Pál; Hajnal, András

**Title: ** On the topological product of discrete \lambda-compact spaces (In English)

**Source: ** General Topology and its Relations to modern Analysis and Algebra, Proc. Sympos. Prague 1961, 148-151 (1962).

**Review: ** [For the entire collection see Zbl 111.35001.]

A topological space __X__ is said to be \kappa-compact if every class *M* of closed subsets of __X__ with void intersection contains a subclass *M*' \subseteq *M* having a void intersection and a power \overline{\overline{*M*'}} with \overline {\overline{*M*'}} < \aleph_{\kappa}.

For each cardinal number m and each pair of ordinal numbers \lambda,\kappa, one uses the abbreviation \top(m,\lambda) ––> \kappa of the statment "if *F* is a class of discrete \lambda-compact topological spaces with the power \overline{\overline*F*} = m then the topological product of the elements of *F* is \kappa-compact". The authors give an outline of the proof of the theorem "if \alpha, \gamma are ordinals such that \aleph_{\alpha+\gamma} is singular and cf(\gamma) < \omega then the statement \top(\aleph_{\alpha+\gamma},\alpha+1) ––> \alpha+\gamma is false" (using the generalized continuum-hypothesis); they discuss some related problems, too. By the theorem the question "\top (\aleph_{\omega},1) ––> \omega?", stated in another paper of the authors [see Acta Math. Acad. Sci. Hung. 12, 87-123 (1961; Zbl 201.32801)], is answered negative.

**Reviewer: ** G.Grimeisen

**Classif.: ** * 54D45 Local compactness, etc.

54A25 Cardinality properties of topological spaces

**Index Words: ** topology

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag