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{\overlineF} = 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
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