Separable space ↩

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created: 2021-12-15 14:31:07
modified: 2022-01-10 04:13:04

Let be a metric space.
is said to be separable if it has a countable subset which is dense in .

Hence if is dense in , then every ball in , no matter how small, will contain points of .

Examples

Real line

The real line is separable
Proof: The set of all rational numbers is countable and is dense in .

Complex plane

The complex plane is separable
Proof: A countable dense subset of is the set of all complex numbers whose real and imaginary parts are both rational.