Does a completely regular space imply the Dirac measure. From wikipedia we have the definition: X is a completely regular space if given any closed set F and any point x that does not belong to F, then there is a continuous function, f, from X to the real line R such that f(x) is 0 and, for every y in F, f(y) is 1. And the Dirac measure is defined by: A Dirac measure is a measure on a set X defined for a given x ∈ X and any set A ⊆ X by δx(A) = 0 for x∉A, and δx(A) = 1 for x∈A. It seems the definition for a completely regular space includes the definition of a Dirac measure. The difference seems to be that the Dirac measure does not involve a continous function, but it does seem as though δx(A) = f(x), where the set A for the Dirac measure seems to be the same thing as the set F in the completely regular space. Both f(x)=δx(A)=0 if x∉F or x∉A and 1 otherwise.