Compactness of Gaussian Random Field

Question

When watching a video on youtube, an instructor says that Gaussian Random Field (GRF) is not compact. I understand that in GRF, we draw sample from function $u(x)$ from, in the case of zero mean $$\mathcal{N} \sim (0, K), \quad K_{i,j} = \exp{\big[-||x_{i}-x_{j}||^{2}/l^{2}\big]},$$ with length paramater $l$. I also know that to be compact, it needs to be bounded and closed. I can see that it is bounded in the domain of $x$, let’s say $x \in [0,1]$, but how can we say about the closed property and conclude that it is not compact?