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.
A compact metric space is second-countable. A separable metric space, e.g., a Polish space, is second-countable. (It is not true that separable first-countable spaces are second-countable; a counterexample is the real line equipped with the half-open or lower limit topology that has as basis the collection of half-open intervals $[a, b)$.)
A topological manifold is second-countable iff it is paracompact and has countably many connected components.
A countable coproduct of second-countable spaces is second-countable.
If $X$ is second-countable and there is an open surjection $f: X \to Y$, then $Y$ is second-countable.
Countable products of second-countable spaces are second-countable. Subspaces of second-countable spaces are second-countable.
For second-countable T_3 spaces $X, Y$, if $X$ is locally compact, then the exponential $Y^X$ is second-countable. Cf. Urysohn metrization theorem and Polish space. I (Todd Trimble) am uncertain to what extent the $T_3$ assumption can be removed.