کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4659957 1344343 2012 19 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Variations of selective separability II: Discrete sets and the influence of convergence and maximality
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات هندسه و توپولوژی
پیش نمایش صفحه اول مقاله
Variations of selective separability II: Discrete sets and the influence of convergence and maximality
چکیده انگلیسی

A space X is called selectively separable (R-separable) if for every sequence of dense subspaces one can pick finite (respectively, one-point) subsets Fn⊂Dn such that ⋃n∈ωFn is dense in X. These properties are much stronger than separability, but are equivalent to it in the presence of certain convergence properties. For example, we show that every Hausdorff separable radial space is R-separable and note that neither separable sequential nor separable Whyburn spaces have to be selectively separable. A space is called d-separable if it has a dense σ-discrete subspace. We call a space X D-separable if for every sequence of dense subspaces one can pick discrete subsets Fn⊂Dn such that ⋃n∈ωFn is dense in X. Although d-separable spaces are often also D-separable (this is the case, for example, with linearly ordered d-separable or stratifiable spaces), we offer three examples of countable non-D-separable spaces. It is known that d-separability is preserved by arbitrary products, and that for every X, the power Xd(X) is d-separable. We show that D-separability is not preserved even by finite products, and that for every infinite X, the power X2d(X) is not D-separable. However, for every X there is a Y such that X×Y is D-separable. Finally, we discuss selective and D-separability in the presence of maximality. For example, we show that (assuming d=c) there exists a maximal regular countable selectively separable space, and that (in ZFC) every maximal countable space is D-separable (while some of those are not selectively separable). However, no maximal space satisfies the natural game-theoretic strengthening of D-separability.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Topology and its Applications - Volume 159, Issue 1, 1 January 2012, Pages 253-271