Separable and first-countable not implies second-countable

Verbal statement
It is possible for a topological space to be both separable and first-countable, but not second-countable.

Separable space
A topological space is termed separable if it has a countable dense subset.

First-countable space
A topological space is termed first-countable if for every point in the space, there is a countable basis at that point.

Second-countable space
A topological space is termed second-countable if it admits a countable basis.

Converse
Any second-countable space is both first-countable and separable.

Sorgenfrey line
The Sorgenfrey line, which is defined as the real numbers given the lower limit topology for the usual ordering, is first-countable and separable but not second-countable.