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Michael space; Lindelöf
We define $\omega ^{\omega }$-directedness, investigate various properties to determine whether they have this property or not, and use our results to obtain easier proofs of theorems due to Laurence and Alster concerning the existence of a Michael space, i.e\. a Lindelöf space whose product with the irrationals is not Lindelöf.
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[L] Laurence L.B.: The influence of a small cardinal on the product of a Lindelöf space and the irrationals. Proc. Amer. Math. Soc. 110 (1990), 535-542. MR 1021211
[M] Michael E.: Paracompactness and the Lindelöf property in finite and countable Cartesian products. Comp. Math. 23 (1971), 199-214. MR 0287502 | Zbl 0216.44304
[vD] van Douwen E.K.: The integers and topology. in Handbook of Set-Theoretic Topology, ed. K. Kunen and J.E. Vaughan, North Holland, Amsterdam, 1984. MR 0776619 | Zbl 0561.54004
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