ABSTRACT.Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture.

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ABSTRACT. Theorem 2.2 lists properties equivalent to left separated spaces in the class of Tl with point-countable bases, with examples preventing plausible additions to this list. For example, X is left iff X is a-weakly separated or X has a closure preserving cover by countable closed sets, but X is left separated does not imply that X is a-discrete. Theorem 2.2 is used to show that the following reflection property holds after properly collapsing a supercompact cardinal to W2: If X is a not a-discrete metric space, then X has a not adiscrete subspace of cardinality less than W2. Similar reflection properties are shown true in some models and false in others.

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