If I understand the question correctly, you only need to check two things in order to apply the Urysohn theorem to this space:

1) The space is regular (it is easily seen to be Hausdorff);

2) The space is second-countable.

An open sets in the space is the union of a usual open set with a subset of the rationals. So a closed set is a usual closed set less a subset of the rationals. If F is such a set then , where is closed in the usual sense and . If then either (in which case you can separate x from F by a usual open set), or (in which case {x} is an open set separating x from F). That proves (1).

For (2), you can take a countable base for the usual topology on and add to it all the singleton sets to get a countable base.