Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-22T20:12:08.816Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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