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Spaces with \(\sigma \)-\(n\)-linked topologies as special subspaces of separable spaces. (English) Zbl 1010.54026
A collection of sets is centered if any finite subcollection has a nonempty intersection; it is \(n\)-linked if any subcollection of at most \(n\) elements has nonempty intersection. A collection is \(\sigma \)-centered or \(\sigma \)-\(n\)-linked, if it is the union of countably many centered, or \(n\)-linked collections, respectively. It is known that a compact Hausdorff space has a \(\sigma \)-centered base if and only if it is separable, and that a Tikhonov space has a \(\sigma \)-centered base if and only if it has a separable Hausdorff compactification [cf. R. Levy and R. H. McDowell, Proc. Am. Math. Soc. 50, 426-430 (1975; Zbl 0313.54025)]. This result makes reasonable the question whether a similar characterization (as certain special subspaces of separable spaces) is possible for the spaces with \(\sigma \)-\(n\)-linked bases. The authors give such a characterization. As a corollary they show that, e.g., a regular space with a \(\sigma \)-linked base has weight at most \(2^{\aleph _0}\), and that there is a bound on the cardinalities of Hausdorff spaces with \(\sigma \)-linked bases.

54D65 Separability of topological spaces
54D70 Base properties of topological spaces
54B10 Product spaces in general topology
54C25 Embedding
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