# separable space

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## separable space

[′sep·rə·bəl ′spās]
(mathematics)
A topological space which has a countable subset that is dense.
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Since C is separable, [X.sub.c] is a separable space. It follows immediately from Proposition 1 in  that N(C) [not equal to] 0.
As we all know the set of polynomials is dense in [H.sup.[infinity].sub.v,0], so that [H.sup.[infinity].sub.v,0] is a separable space. In particular, for [alpha] > 0 and v(z) = [(1 - [absolute value of (z)].sup.2]).sup.[alpha]], we obtain [H.sup.[infinity].sub.[alpha],0] and [H.sup.[infinity].sub.[alpha],0].
Let X be a separable space, A a closed subset of X and [??]: [OMEGA] x X [??] X a measurable map with nonempty closed values.
Now, if we assume that, for almost every [omega] [member of] [OMEGA], the Lefschetz set [LAMBDA]([[??].sub.[omega]]) is different from {0} and, in view of the assumption that X is a separable space, we can apply Lemma 3.6 (see also Remark 3.7).
(Open Problem) Is Theorem 5.3 true without the assumption that X is a separable space?
Then [f.sup.-1](K) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a hereditarily separable space, it implies that f -1(K) is separable in M, i.e, f is a cs-map.
However, for separable spaces X it is so : separating subspaces of NA(X) are actually isometric preduals of X and thus in particular they are 1-norming.

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