Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.
Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates.
For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable.
For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.
[1] Therefore, the lower limit topology on the real line is not metrizable.
Urysohn's metrization theorem states that every second-countable, Hausdorff regular space is metrizable.
Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.