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The almost Lindelöf degree. (English) Zbl 0551.54003

In Dokl. Akad. Nauk SSSR 187, 967-970 (1969; Zbl 0191.209) A. V. Arkhangel’skij showed that for any \(T_ 2\) space X, \(| X| \leq 2^{L(X)\chi(X)},\) where \(L(X)\) is the Lindelöf degree of \(X\) and \(\chi(X)\) is the character of X. In Pac. J. Math. 79, 37-45 (1978; 367.54003) M. Bell, J. Ginsburg and G. Woods improved this result, assuming normality, by showing that for \(T_ 4\) spaces X,\(| X| \leq 2^{wL(X)\chi(X)},\) where wL(X) is the weak Lindelöf degree of \(X\). We introduce a new cardinal function aL(X), the almost Lindelöf degree of \(X\), which agrees with \(L(X)\) on \(T_ 3\) spaces, but which is often smaller than L(X) on \(T_ 2\) spaces, and show that for \(T_ 2\) spaces X, \([X]\leq 2^{aL(X)\chi(X)}.\)

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)

Citations:

Zbl 0191.209
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