CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A space (such as a topological space) is second-countable if, in a certain sense, there is only a countable amount of information globally in its topology. (Change ‘globally’ to ‘locally’ to get a first-countable space.)
A topological space is second-countable if it has a countable basis $B$.
A locale is second-countable if there is a countable set $B$ of open subspaces (elements of the frame of opens) such that every open $G$ is a join of some subset of $B$. That is, we have
The weight of a space is the minimum of the cardinalities of the possible bases $B$. We are implicitly using the axiom of choice here, to suppose that this set of cardinalities (which really is a small set because bounded above by the number of open subspaces, and inhabited by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.
Then a second-countable space is simply one with a countable weight.
Any space first-countable space must be second-countable. (Conversely, any second-countable space must be first-countable.) In particular, separable metric spaces are second-countable.
A topological manifold is second-countable iff it is paracompact and has countably many connected components.