Abstract
We prove that the cardinals μ which may be the number of ideals of an infinite Boolean algebras are restricted: \(\mu = \mu ^{\aleph _0 }\) and if κ≤μ is strong limit then μ<κ=μ. Similar results hold for the number of open sets of a compact space (we need w(x) <ŝ(x)=2<ŝ(x)). We also prove that if μ≥⊃2 is the number of open subsets of a Hausdorff space X,\(\mu < \mu ^{\aleph _0 }\) then 0# exists, (in fact, the consequences of the covering lemma on cardinal arithmetic are violated). We also prove that if the spread μ of a Hausdorff space X satisfies μ>⊃2(c f μ) that the sup is obtained. For regular spaces μ;>2cf μ is enough.
Similarly for 3(X) and h (X).
Keywords
- Open Subset
- Topological Space
- Boolean Algebra
- Strong Limit
- Hausdorff Space
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© 1986 Springer-Verlag
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Shelah, S. (1986). Remarks on the numbers of ideals of Boolean algebra and open sets of a topology. In: Around Classification Theory of Models. Lecture Notes in Mathematics, vol 1182. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098509
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DOI: https://doi.org/10.1007/BFb0098509
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