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Variations of selective separability II: discrete sets and the influence of convergence and maximality | | Angelo Bella
; Mikhail Matveev
; Santi Spadaro
; | | Date: |
24 Jan 2011 | | Abstract: | A space $X$ is called selectively separable(R-separable) if for every
sequence of dense subspaces $(D_n : ninomega)$ one can pick finite
(respectively, one-point) subsets $F_nsubset D_n$ such that
$igcup_{ninomega}F_n$ 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
emph{d-separable} if it has a dense $sigma$-discrete subspace. We call a
space $X$ D-separable if for every sequence of dense subspaces $(D_n :
ninomega)$ one can pick discrete subsets $F_nsubset D_n$ such that
$igcup_{ninomega}F_n$ 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 $X^{2^{d(X)}}$ is not
D-separable. However, for every $X$ there is a $Y$ such that $X imes Y$ is
D-separable. Finally, we discuss selective and D-separability in the presence
of maximality. For example, we show that (assuming ${mathfrak d}=mathfrak c$)
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. | | Source: | arXiv, 1101.4615 | | Services: | Forum | Review | PDF | Favorites | |
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