A space X is called selectively separable (R-separable) if for every sequence of dense subspaces (Dn : n ∈ ω) one can pick finite (respectively, onepoint) subsets Fn ⊂ Dn such that n∈ω Fn 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 (Dn : n ∈ ω) one can pick discrete subsets Fn ⊂ Dn such that n∈ω Fn is dense in X. Although dseparable 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 × 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.2000 Mathematics Subject Classification. 54D65, 54A25, 54D55, 54A20.

A space is said to be almost discretely Lindelöf if every discrete subset can be covered by a Lindelöf subspace. In [10], Juhász, Tkachuk and Wilson asked whether every almost discretely Lindelöf first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under 2

We prove upper bounds for the spread, the Lindelöf number and the weak Lindelöf number of the G δ topology on a topological space and apply a few of our bounds to give a short proof to a recent result of Juhász and van Mill regarding the cardinality of a σ-countably tight homogeneous compactum.2000 Mathematics Subject Classification. Primary: 54A25, Secondary: 54D20, 54G20.

We present a bound for the weak Lindelöf number of the G δ -modification of a Hausdorff space which implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: |X| ≤ 2 L(X)χ(X) (Arhangel'skiȋ) and |X| ≤ 2 c(X)χ(X) (Hajnal-Juhász). This solves a question that goes back to Bell, Ginsburg and Woods [6] and is mentioned in Hodel's survey on Arhangel'skiȋ's Theorem [15]. In contrast to previous attempts we do not need any separation axiom beyond T 2 .