Remark

If A is a dense subset of \mathbf{R}, the topology of \mathbf{R} is generated by the open intervals whose end-points belong to A. For if

is an open interval containing x, there exist two points y, z of A such that x - a < y < x and x < z < x + a; hence ]y, z[ contains x and is contained in ]x - a, x + a[. This proof shows, moreover, that the intervals under consideration form a base (Chapter I, § 1, no. 3) of the topology of \mathbf{R}. In particular, if we take A = \mathbf{Q}, we see that the topology of \mathbf{R} has a countable base.