Second Countable Spaces, and the relation to First Countable Spaces

A topological space (X, T) is second countable if the topology T has a countable basis. The real line with the usual topology is second countable, whereas the real line with the lower-limit topology is not second countable. We also show that second countable spaces are also first countable. The converse is not true, which we demonstrate using the real line with the lower limit topology.