While preparing a lecture on dynamic programming principle in optimal stochastic control after the book of Touzi, I discovered a gap in the proof of DPP (page 28 of the book).

Here I simplify the relevant place to give a brief idea where the gap lies. The author defines a family of sets $$ B(s,y) = \{(t,x)\in \mathbf{S}=[0,T)\times \mathbb{R}^d: t\in (s-r(s,y),s), |x-y|<r(s,y)\}, $$ where $(s,y) \in \mathbf{S}$, and $r(s,y)>0$ are some numbers. Further, the author writes, "Clearly, $\{B(s,y): (s,y) \in \mathbf{S}\}$ forms an open covering of $\mathbf{S}$", and then appeals to Lindelöf's theorem to show that there is a countable subcover. However, the fact that $\{B(s,y)\}$ is a cover is false in general.

I've managed to find a workaround. Instead of considering the open cylinders $B(s,y)$, one needs to consider half-open cylinders $$ D(s,y;r) = \{(t,x)\in \mathbf{S}: t\in (s-r,s], |x-y|<r\} $$ (note that $t=s$ is now included to the set). Then we need the following.

Fact 1Let, for each $r(s,y)$, $(s,y) \in \mathbf{S}$, be arbitrary positive numbers. Then there is a countable family $\{(s_n,y_n),n\ge 1\}$ such that $$ \mathbf{S}= \bigcup_{n\ge 1} D(s_n,y_n; r(s_n,y_n)). $$

The proof is as follows. Obviously, $\mathsf{D} = \{D(s,y,r(s,y)),(s,y) \in \mathbf{S}\}$ is a cover of $\mathbf{S}$. The sets $D(s,y;r)$ are open in the topology of the product space $[0,T)\times \mathbb{R}^d$, where the interval $[0,T)$ is equipped with the left half-open interval topology, and $\mathbb{R}^d$ is equipped with the usual topology. Since this space is a product of a Lindelöf space and a $\sigma$-compact space, it is Lindelöf itself. Therefore, there exists a countable subcover of $\mathsf{D}$.

I don't like this argument too much, since it involves some topological references I would like to avoid. Therefore, the question:

Is there a simpler/direct argument of Fact 1?

Fact 1should read $$\mathbf{S}\subset \bigcup_{n\ge 1} D(s_n,y_n; r(s_n,y_n)) . $$ $\endgroup$Sshould belong to cylinders that also include points $(-\epsilon,x)$. $\endgroup$