On the weight-spectrum of a compact space
The weight-spectrumSp(w, X) of a spaceX is the set of weights of all infinite closed subspaces ofX. We prove that ifκ>ω is regular andX is compactT 2 withω(X)≥κ then some λ withκ≤λ≤2<κ is inSp(ω, X). Under CH this implies that the weight spectrum of a compact space can not omitω 1, and thus solves problem 22 of [M]. Also, it is consistent with 2ω=c being anything it can be that every countable closed setT of cardinals less thanc withω ∈ T satisfiesSp(w, X)=T for some separable compact LOTSX. This shows the independence from ZFC of a conjecture made in [AT].
KeywordsCompact Space Closed Subspace Inaccessible Cardinal Cardinal Function Singular Cardinal
Unable to display preview. Download preview PDF.
- [ASh]A.V. Arhangel’skii and B.E. Shapirovskii,The structure of monolithic spaces, Vestnik. Mosk. Univ., Ser. I4 (1987), 72–74.Google Scholar
- [vD]E. van Douwen,Cardinal functions on compact F-spaces and on weakly countably complete Boolean algebras, Fund. Math.114 (1981), 236–256.Google Scholar
- [HJ3]A. Hajnal and I. Juhász,A consequence of Martin’s axiom, Indag. Math.33 (1971), 457–463.Google Scholar
- [J1]I. Juhász,Cardinal functions — ten years later, Math. Centre Tract no. 123, Amsterdam, 1980.Google Scholar
- [J2]I. Juhász,Two set-theoretic problems in topology, Proc. 4th Prague Symp. on Gen. Top. (1976), Part A, pp. 115–123.Google Scholar
- [JSz]I. Juhász and Z. Szentmiklóssy,On convergent free sequences in compact spaces, Proc. Amer. Math. Soc., to appear.Google Scholar
- [K]K. Kunen,Set Theory, Studies in Logic, Vol. 102, North-Holland, Amsterdam, 1980.Google Scholar
- [W]W. Weiss,Versions of Martin’s axiom, inHandbook of Set-theoretic Topology (R. Kunen and J. Vaughan, eds.), North-Holland, Amsterdam, 1989, pp. 827–886.Google Scholar