Suppose that $mathbb E:={E_r(x)}_{rin mathcal I, xin X}$ is a family of open subsets of a topological space $X$ endowed with a nonnegative Borel measure $mu$ satisfying certain basic conditions. We establish an $mathcal{A}_{mathbb E, p}$ weights theory with respect to $mathbb E$ and get the characterization of weighted weak type (1,1) and strong type $(p,p)$, $1<ple infty$, for the maximal operator $mathcal M_{mathbb E}$ associated with $mathbb E$. As applications, we introduce the weighted atomic Hardy space $H^1_{mathbb E, w}$ and its dual $BMO_{mathbb E,w}$, and give a maximal function characterization of $H^1_{mathbb E,w}$. Our results generalize several well-known results.