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Variations of selective separability Ⅱ: Discrete sets and the influence of convergence and maximality

机译：选择性可分离性的变化Ⅱ：离散集及其收敛性和最大值的影响

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摘要

A space X is called selectively separable (R-separable) if for every sequence of dense subspaces (D_n: n ∈ω) one can pick finite (respectively, one-point) subsets F_n (∩) D_n such that ∪_(n∈ω) F_n 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 (D_n: n∈ω) one can pick discrete subsets F_n (∩) D_n such that ∪_(n∈ω) F_n 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 x~(d(x)) is d-separable. We show that D-separability is not preserved even by finite products, and that for every infinite X, the power X~(2d(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 σ = 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.
机译：如果对于每个密集子空间序列（D_n：n∈ω）可以选择有限的（分别为单点）子集F_n（∩）D_n，使得∪_（n∈ ω）F_n在X中是密集的。这些属性比可分离性要强得多，但是在存在某些会聚属性时等效于它。例如，我们表明每个Hausdorff可分离的径向空间都是R可分离的，并且请注意，可分离的连续空间或可分离Whyburn空间都不必须是可选择性分离的。如果空间具有密集的σ离散子空间，则称为d可分离的空间。如果对于每个密集子空间序列（D_n：n∈ω）可以选择离散子集F_n（∩）D_n，使得∪_（n∈ω）F_n在X中是稠密的，则我们称空间X D可分离。可分离空间通常也是D可分离的（例如，对于线性有序d可分离或可分层空间），我们提供了三个可数不可D可分离空间的示例。众所周知，d的可分离性可以通过任意乘积来保持，并且对于每个X，幂x〜（d（x））是d可分离的。我们表明，即使有限乘积也不能保持D可分离性，并且对于每个无限X，幂X〜（2d（x））都不是D可分离的。但是，对于每个X，都有一个Y，使得X×Y是D可分离的。最后，我们讨论在最大存在下的选择性和D可分离性。例如，我们证明（假设σ= c）存在一个最大的规则可数选择性分离的空间，并且（在ZFC中）每个最大可数空间都是D可分离的（而其中某些不是选择性可分离的）。但是，没有最大的空间可以满足D可分离性自然博弈论的要求。

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