A separable metric space is completely metrizable if and only if the second player has a winning strategy in this game.

For separable spaces, the notions of weak and strong measurability agree.

Let X be an arbitrary separable metric space.

The two notions above agree for separable, metrisable spaces.

Now let be a complete separable metric space, the phase space.

Any continuous image of a separable space is separable .

A product of at most continuum many separable spaces is separable.

The notions of regularity and completeness are incompatible in a separable space.

With respect to either σ or σ, D is a separable space.

R is not metrizable, since separable metric spaces are second-countable.