# Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space)

"Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed under the operations $\vee, \wedge$ contains the function $1$ and is dense in C(K)"

I can't see how to construct such a set $H$ or stated differently: Why can I assume the existence of such a set?

Remark:

Well the passage is from a book of Dellacherie on Stochastic Processes. It is used in a proof of the Disintegration property of Measures and it truly is used as a fact and not as an assumption. That is, the proof of the theorem begins by stating "Let H...." and this is not part of any assumption underlying the statement of the theorem. (i don't think it is helpful - but the theorem states that on a compact metric space with its Borel sigma field the disintegration property holds)

• What are the operations $\vee ,\wedge$? – MotylaNogaTomkaMazura Jun 30 '15 at 8:31
• Probably $\min(f,g)$ and $max(f,g)$, but I may be wrong – Tryss Jun 30 '15 at 8:34