Existence of Separable Space which is not Second-Countable

Theorem

There exists at least one example of a separable topological space which is not also a second-countable space.

Proof

Let $T$ be the Sorgenfrey line.

From Sorgenfrey Line is Separable, $T$ is a separable space.

From Sorgenfrey Line is not Second-Countable, $T$ is not a second-countable space.

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Countability Axioms and Separability