Functionally Hausdorff space

Definition
A topological space is termed a completely Hausdorff space or functionally Hausdorff space if it satisfies the following equivalent conditions:


 * 1) For any two points in it, there is a continuous function from the whole space to $$[0,1]$$ that takes the value $$0$$ at one point and $$1$$ at the other.
 * 2) For any two points in it, there is a continuous function from the whole space to the reals that takes the value $$0$$ at one point and $$1$$ at the other.
 * 3) For any two points in it, there is a continuous function from the whole space to the reals that takes distinct values at the two points.
 * 4) For any two points in it, and any two specified distinct real numbers, there is a continuous function from the whole space to the reals that takes the two specified values at the two points.

Facts
Any connected Urysohn space with at least two points is uncountable (more precisely, its cardinality must be at least that of the continuum). This follows from the fact that its image under any continuous function must be connected, and hence the Urysohn function separating two points must be surjective to $$[0,1]$$.