I have a question on the sample path right-continuity of a stochastic process.
Let $X=\{X_t\}_{t \ge 0}$ be a stochastic process on a locally compact separable metric space $E$. We assume that $X$ starts from $x \in E$ and the sample path of $X$ is right continuous on $[0,\infty)$ with left limits in $E$.
Let $A $ be an open subset of $E$. We set \begin{align*} \sigma_{A}=\inf\{t>0 \mid X_t \in A\}. \end{align*} We consider the following event. For $t>0$, \begin{align*} \{X_{kt/2^n}\in E \setminus A \mid k=1,2,\ldots,2^n,\,\,n=1,2,\ldots\} \end{align*}
it clearly holds that $X_t \in E \setminus A$. This, I think the above event is equal to $\{t<\sigma_A\}$. Is this wrong?