# Borel $\sigma$-algebra on $C(T)$, a set of continuous functions on a compact pseudo-metric space $T$.

I study probability theory, and I'm stuck on the problem below.

$$T$$ : compact pseudo-metric space ($$d(x,y)=0 \nRightarrow x=y$$)
$$S$$ : dense subset of $$T$$
$$C(T)$$ : a set of all continuous functions on $$T$$
A compact pseudo-metric space is separable.

I know the Borel $$\sigma$$-algebra on $$C(T)$$ is the smallest $$\sigma$$-algebra making all projections $$z\mapsto z(s)$$ measurable ($$s\in S,z\in C(T)$$), but I would like to ask you how to prove it.

There is a hint.
First, prove the closed ball around $$z_0$$ of radius $$r$$ equals $$\bigcup_{s\in S_0}\{z:|z(s)-z(s_0)|\leq r\}$$ , where $$S_0$$ is a countable dense subset of $$S$$ .
Then, prove every closed and open ball is projection measurable .
Finally, prove the statement of this proposition.

• Where exactly did you get stuck? Did you justify the first part of the hint? – Kavi Rama Murthy Sep 16 at 12:49
• I tried, but did not justify any part of them. – Joey Sep 16 at 12:54
• @KaviRamaMurthy So first part . – Joey Sep 17 at 7:50