First Examples
Any topological space which is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset. Similarly the set of all vectors in which is rational for all i is a countable dense subset of ; so for every the -dimensional Euclidean space is separable.
A simple example of a space which is not separable is a discrete space of uncountable cardinality.
Further examples are given below.
Read more about this topic: Separable Space
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