Let X be a completely regular, Hausdorff space in the following. A subset S of x is said to be \(T_ z\)-embedded (totally z-embedded) in X if every disjoint family of cozero sets of S extends to a disjoint family of cozero sets of X. Every \(T_ z\)-embedded subset is z-embedded, and the concepts are identical for dense subsets. The following are shown to be equivalent: (a) every open subset of X is \(T_ z\)-embedded, (b) every dense subset of X is \(T_ z\)-embedded, (c) every open subset of X is z- embedded. Every subset of X is \(T_ z\)-embedded iff X is a hereditarily collectionwise normal space for which every open subset is z-embedded. The space X is \(T_ z\)-embedded in every completely regular, Hausdorff space in which it is embedded iff X satisfies the countable chain condition and is almost compact or Lindelöf. The analogous properties of \(T_ G\)-, \(D_ u\)- and \(D_ G\)-embeddedness are also discussed.