Are Borel probability measures on second countable, locally compact, Hausdorff spaces automatically regular?

Question

This is a quote from the wikipedia entry on regular measures:

Any Borel probability measure on a locally compact Hausdorff space with a countable base for its topology ... is regular.

Is this actually true? If so could someone point me to a reference where I could find a proof of the statement?

Edit: I should add that the definition of regularity used in wikipedia is that $\mu$ is regular if

$$\mu(A)=\sup\{\mu(F):F\subset A\text{ and $F$ is compact}\}=\inf\{\mu(B):A\subset B\text{ and $B$ is open}\}$$

where the infimum/supremum is taken over the space's Borel sigma algebra.

Answer 1

Proposition 7.2.3. in Measure Theory by Donald L. Cohn.