Therefore M is norm separable and a fortiori [OMEGA]* (as a subspace of a

separable metric space).

Let S be a

separable metric space, X be a real Banach space and G : S [right arrow] P([L.sup.1](I, X)) be a lower semicontinuous set-valued map with closed decomposable values.

Theorem 3.6 Any

separable metric space admits an isometric embedding into Q.

(3) X is a compact-covering image of a

separable metric space,

To make this question precise, we must consider the distal flows as points in a

separable metric space. (For example we can consider the generator of a Z-flow as a compact subset of the square of the Hilbert cube.) We discuss several ways of doing this and show that they are all equivalent up to Borel reductions.

Let ([OMEGA], [summation]) be a measurable space and Y be a

separable metric space. A map [??]: [OMEGA] [??] Y with closed values is called measurable if [[??].sup.-1](B) [member of] [summation], for each open B [subset] Y, or equivalently, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [member of] [summation], for each closed B [subset] Y.

Using the terminology of the previous section, this space is obtained by taking the free Banach space over a particular complete

separable metric space known as the Urysohn space.

(2) X is a subsequence-covering (resp., sequence-covering, 1-sequence-covering image of a

separable metric space.

We asserted that if X is a connected

separable metric space such that [[pi].sup.top.sub.n](X, x) is discrete, then [[pi].sub.n](X, x) is countable, and as a result we showed that if X is a connected locally n-connected

separable metric space, then [[pi].sub.n](X, x) is countable.

(3) X is a quotient mssc-image of a

separable metric space.

Hattori, "Dimension and superposition of bounded continuous functions on locally compact,

separable metric spaces," Topology and its Applications, vol.

Gregoriades examines the equivalence classes under effective Borel isomorphism, between complete

separable metric spaces that admit a recursive presentation, and show the existence of strictly increasing and strictly decreasing sequences as well as of infinite anti-chains under the natural notion of effective isomorphism reduction, as opposed to the non-effective case, where only two such classes exist, the one of the Baire space and the one of the naturals.