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First countable spaces without point-countable π-bases

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (2007)

Fundamenta Mathematicae

We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that ∙ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable π-base (in fact, the minimum order of a π-base of the space can be made arbitrarily large); ∙ if there is a κ-Suslin line then there is a first countable GO-space of cardinality κ⁺ in which the order of any π-base is at least κ; ∙ it is consistent to have a first countable,...

Forcing tightness in products of fans

Jörg Brendle, Tim La Berge (1996)

Fundamenta Mathematicae

We prove two theorems that characterize tightness in certain products of fans in terms of families of integer-valued functions. We also define several notions of forcing that allow us to manipulate the structure of the set of functions from some cardinal θ to ω, and hence, the tightness of these products. These results give new constructions of first countable <θ-cwH spaces that are not ≤θ-cwH.

Further remarks on KC and related spaces

Angelo Bella, Camillo Costantini (2011)

Commentationes Mathematicae Universitatis Carolinae

A topological space is KC when every compact set is closed and SC when every convergent sequence together with its limit is closed. We present a complete description of KC-closed, SC-closed and SC minimal spaces. We also discuss the behaviour of the finite derived set property in these classes.

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