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A new bound on the cardinality of homogeneous compacta. (English) Zbl 1098.54002

The author provides new bounds on the cardinality and weight of compact homogeneous spaces in terms of the tightness of the space. The main result proved is that \(w(X)\leq 2^{t(X)}\) for a compact homogeneous space \(X\). From this result the author deduces the following 1) the cardinality of a compact homogeneous \(X\) is bounded by \(2^{t(X)}\), so in particular, 2) compact homogeneous spaces of countable tightness have cardinality at most continuum (answering a question of Arhangel’skii). 3) Assuming GCH, the tightness and character of compact homogeneous spaces coincide. 4) Assuming \(2^{\aleph_0}<2^{\aleph_1}\), homogeneous \(T_5\) compact spaces are first countable.

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E35 Consistency and independence results
54D30 Compactness
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