Separable Space - Properties

Properties

  • A subspace of a separable space need not be separable (see the Sorgenfrey plane and the Moore plane), but every open subspace of a separable space is separable, (Willard 1970, Th 16.4b). Also every subspace of a separable metric space is separable.
  • In fact, every topological space is a subspace of a separable space of the same cardinality. A construction adding at most countably many points is given in (Sierpinski 1952, p. 49).
  • The set of all real-valued continuous functions on a separable space has a cardinality less than or equal to c. This follows since such functions are determined by their values on dense subsets.
  • From the above property, one can deduce the following: If X is a separable space having an uncountable closed discrete subspace, then X cannot be normal. This shows that the Sorgenfrey plane is not normal.
  • For a compact Hausdorff space X, the following are equivalent:
(i) X is second countable.
(ii) The space of continuous real-valued functions on X with the supremum norm is separable.
(iii) X is metrizable.

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