Editing Separable space (section)

==Separability versus second countability==
Any [[second-countable space]] is separable: if <math>\{U_n\}</math> is a countable base, choosing any <math>x_n \in U_n</math> from the non-empty <math>U_n</math> gives a countable dense subset. Conversely, a [[metrizable space]] is separable if and only if it is second countable, which is the case if and only if it is [[Lindelöf space|Lindelöf]].

To further compare these two properties:
* An arbitrary [[subspace (topology)|subspace]] of a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).
* Any continuous image of a separable space is separable {{harv|Willard|1970|loc=Th. 16.4a}}; even a [[quotient topology|quotient]] of a second-countable space need not be second countable.
* A [[product topology|product]] of at most continuum many separable spaces is separable {{harv | Willard | 1970 | loc=Th 16.4c | p=109 }}. A countable product of second-countable spaces is second countable, but an uncountable product of second-countable spaces need not even be first countable.

We can construct an example of a separable topological space that is not second countable.  Consider any uncountable set <math>X</math>, pick some <math>x_0 \in X</math>, and define the topology to be the collection of all sets that contain <math>x_0</math> (or are empty).  Then, the closure of <math>{x_0}</math> is the whole space (<math>X</math> is the smallest closed set containing <math>x_0</math>), but every set of the form <math>\{x_0, x\}</math> is open.  Therefore, the space is separable but there cannot have a countable base.