Let $G$ be a topological group and $H$ is a subgroup of $G$. Suppose that the subspace topology on $H$ is the discrete topology. Prove that the quotient map $P:G \to G/H$ is a covering space.
My attempt:
For an element $a_1H$ in $G/H$ open nbd $U$ of $a_1H$ in $G/H$ then $p^{-1}(U)$ contains the elements of the type $ah$ now how can I divide it into open disjoint slices? For bijectiveness I think $a_i$ s.t $i \in I$ index set where each $a_i$ generate different $a_iH \in U$ . But why this is open in $G$?