separable space
A separable space is one you can survey with a countable list of sample points so fine that nothing in the space is far from the list. Despite possibly containing uncountably many points, the space can be approximated to any precision using only a sequence of representatives.
Formally, a metric space is separable if it has a countable dense subset — a subset that is both countable and whose closure is the whole space. Separability is a smallness or manageability condition: it lets many constructions proceed one approximating point at a time, and it is enjoyed by essentially all spaces that arise in everyday analysis.
The real line R is separable because the rationals Q are a countable dense subset, and Euclidean R^n is separable for the same reason using points with rational coordinates. Most function spaces of practical interest are separable as well, though not all: the space of bounded sequences with the sup metric is a standard example of a metric space that is not separable.
R^2 is separable: the countable set of points with both coordinates rational, like (3/4, -2/5), is dense, since any point can be approximated as closely as desired by such a point.
Rational-coordinate points form a countable dense grid in the plane.