Countable subcover of half-open cylinders

Question

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 1 Let, 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?

Answer 1

Let $B_r$ denote the closed ball of radius $r$ centered at $0$ in $\mathbb{R}^d$.

Claim: for any $r>0$ and $\tau <T$ there is a countable subfamily of $D$ that covers $[0,\tau] \times B_r$.

To prove the claim, consider the set $I$ of all $t\in [-\infty, \tau]$ such that $(t, \tau] \times B_r$ is covered by some countable subfamily of $\mathsf{D}$. The set $I$ is not empty (trivially, $\tau$ is in $I$); it is closed, for if $t_j$ is a sequence in $I$ converging to $t$, then also $(t,\tau] \times B_r$ is covered by a countable subfamily of $\mathsf{D}$ (namely, by the countable union of the countable families that covers the sets $(t_j,\tau] \times B_r$, for $j$ in $\mathbb{N}$). Next, we show that $\min I < 0$, which implies the claim.

Notice that for any $0 \le t \le \tau$ there is a finite subfamily of $\mathsf{D}$ which covers the set $\{t \}\times B_r$. Indeed, the traces of the cylinders $D \in \mathsf{D}$ on the compact subspace $\{t \}\times B_r$ are an open ball covering of it. Also note that this finite family of balls also covers $(t',t]\times B_r$ for some $t'<t$. Therefore it can't be the case that $\min I \ge 0 $.

Since for any rational $r>0$ and $\tau<T$ there is a countable subfamily of $D$ that covers of $[0,\tau] \times B_r$, the union of these is a countable covering of $[0,T) \times \mathbb{R}^d$.