Let $X_{t\in T}$ be a random function where $T$ is a subset of $\mathbb{R}^n$.

Since $T$ has inifite points, we are not able to use union bound to estimate $\sup X_t$. Thus instead, when $T$ is compact (then the covering number of $T<+\infty$), we could use $\epsilon$-net argument to derive a meaningful upper bound on $P(\sup X_t\geq\cdots)\leq$.

However in my case, $T$ is a rectangle with one open line, i.e. $[0,1]^n\times (0,1]$. I was wondering could we use $\epsilon$-net in this case?

Many thanks for suggestions!