I am reading some paper about stochastic processes, but I am not familiar with measure theory. The paper assumed a locally compact and separable metric space $X$ and the Borel field $\mathcal{B}(X)$ on $X$.
I am trying to use simple examples to understand these concepts. Consider $X=\mathbb{R}_{\ge0}$ and $\tau=\{\varnothing, A, A^c, B, B^c, A\cup B, A^c\cap B^c, \mathbb{R}_{\geq0}\}$ where $A=[1, 2]$ and $B=[3,\infty)$. Is $(X,\tau)$ a topological space and then also a locally compact and separable metric space? If that, does it mean $\mathcal{B}(X)=\{\varnothing, A, A^c, B, B^c, A\cup B, A^c\cap B^c, \mathbb{R}_{\geq0}\}$?
I'll leave it to you to check that your $\tau$ is closed under unions and intersections and hence a topology (in general, I should say finite intersections but your $\tau$ is finite). It cannot be induced by a metric since it is not Hausdorff: any open set that contains $1/3$ also contains $2/3$. $\tau$ is locally compact because any finite topology is automatically locally compact: any covering of any subset is finite and hence every subset is compact. In particular, all open neighbourhoods are compact.
If you want examples of locally compact, separable metric space where you can enumerate all the sets of interest, take $X$ to be finite and give it the discrete topology. As suggested in the comments, intervals of $\Bbb{R}$ under the standard metric also provide important examples, but then you have to deal with an infinite collection of open sets.