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Martin's axiom and a regular topological space with uncountable net weight whose countable product is hereditarily separable and hereditarily Lindelöf

Published online by Cambridge University Press:  12 March 2014

Krzysztof Ciesielski
Krzysztof Ciesielski*
Affiliation:
Department of Mathematics, Bowling Green State University, Bowling Green, Ohio 43403 Department of Mathematics, Warsaw University, Warsaw, Poland
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Extract

In [1, p. 51] A. V. Arhangel'skiĭ, in connection with the problems of L-spaces and S-spaces, examined further the notions of hereditary separability and hereditary Lindelöfness. In particular he considered the following property P: “Every regular topological space has a countable net weight provided its countable product is hereditarily Lindelöf and hereditarily separable.” He noticed that the continuum hypothesis implies negation of the property P and posed a question: “Do Martin's Axiom and the negation of the continuum hypothesis imply P?” The purpose of this paper is to give a negative answer to this question.

The set-theoretical and topological notation that we use is standard and can be found in [6] and [5] respectively.

Throughout the paper we will use the notation H(X, Y) to denote the set of all finite functions from a set X to Y.

Theorem. Con(ZFC) → Con(ZFC + MA + ¬CH + there exists a 0-dimensional Hausdorff space X such that nw(X) = с and nw(Y) = ω for any Y ϵ [X]<с).

Proof. Let M be a model of ZFC satisfying CH and let F be an M-generic filter over the Cohen forcing {H(ω2 × ω2, 2), ⊃). Then f = ⋃F is a function and f: ω2 × ω2 → 2.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 2 , June 1987 , pp. 396 - 399
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[1]Arhangel'skiĭ, A. V., The structure and the classification of topological spaces and cardinal invariants, Uspehi Matematičeskih Nauk, vol. 33 (1978), no. 6 (204), pp. 29–83 (Russian); English translation, Russian Mathematical Surveys, vol. 33 (1978), no. 6, pp. 33–96.Google Scholar
[2]Burgess, J. P., Forcing, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 403–453.CrossRefGoogle Scholar
[3]Ciesielski, K., On the netweight of subspaces, Fundament a Mathematicae, vol. 117 (1983), pp. 37–46.CrossRefGoogle Scholar
[4]Ciesielski, K., The topologies generated by graphs, Proceedings of the Jadwisin conference 1981, University of Leeds Press, Leeds, 1983, pp. 67–92.Google Scholar
[5]Juhász, I., Cardinal functions in topology—ten years later, Mathematisch Centrum, Amsterdam, 1980.Google Scholar
[6]Kunen, K., Set theory, North-Holland, Amsterdam, 1980.Google Scholar