I am trying to understand the proof of Theorem 6.15 in Le Gall's book Brownian Motion, Martingales, and Stochastic Calculus about the regularity of sample paths of Markov processes (that they have a càdlàg modification) which is also Markov with respect to the completed filtration.
However, I am having difficulty understanding some separability statements related to point-set topology.
The topological space $E$ under consideration is assumed to be metrizable, locally compact, and $\sigma$-compact.
In a step of the proof, Le Gall states that we can find a sequence of nonnegative bounded functions (let the space denoted by $C^+_0(E)$) which separates the points of $E_{\Delta}$, which is the Alexandroff compactification of $E$.
He then states that the subset $\mathcal{H} = \{R_pf_n: p \geq 1, n \geq 0\}$ is a countable subset of $C^+_0(E)$ and separates the points of $E_{\Delta}$, where $R_p$ is the resolvent operator $R_pf(x) = \int_{0}^{\infty} e^{pt} Q_t f(x) dt$, and $(Q_t)_{t \geq 0}$ is a transitional semigroup (in this theorem a Feller semigroup).
I can understand why there exists such a sequence (because $E$ is metrizable thus normal, and appeal to the Urysohn lemma), I can also understand why $\mathcal{H}$ separates the points of $E_{\Delta}$, but I am having some trouble seeing why such a set indexed by positive real numbers $p$ is countable.
My question is: why is $\mathcal H$ countable? Or am I understanding something incorrectly about the definitions?
For your reference, here is a screenshot of the theorem and the part of the proof I have problems with:
He certainly means $p \in \mathbb{N} = \mathbb{N}_{\ge1 }$.
The next sentence even is "If $h \in \mathscr{H}$ , Lemma 6.6 shows that there exists an integer $p \geq 1$ ...".