question about independence of a random variable and a sequence of random variables

Question

Let $X$ and $Z_n, n \in \mathbb{N}$ be real random variables defined on the same probability space. Does $X$ is independet of $(Z_n)_{n \in \mathbb{N}}$ mean that the sigma-algebras $\sigma(X)$ and $\sigma(Z_n)$ are pairwise independent for all $n$ or that $\sigma(X)$ and $\sigma(Z_n:n\in \mathbb{N})$ are independent?