Given a set $X$ there are two ways to turn it into a topological space, first, specify the convergence (of nets), second, specify the topology. The two ways are equivalent.
Let $X=C(0,1)$ equipped with the $L^p$ norm, $p<\infty$. This specifies a topology on $X$. Obviously, $X$ is closed, because this is so in every topology. However, $X$ is incomplete. In fact, the completion of $X$ in $L^p$ norm is the $L^p(0,1)$.
Suppose that we have completed $X$ to $\overline X=L^p$. How did the act of completion change the topology on $X$? In the $L^p$ topology, we have a relative topology on $X$. Is this the same as the topology that we started with? I think no. This is because $X$ as a subset of $L^p$ is NOT closed in the relative topology, even though $X$ was closed before the completion.
As such, the act of completion throws away some open sets. But, which ones? And what exactly did it do?
Completion doesn't do anything to the topology. It doesn't matter whether $X$ is closed in the completion or not; the subspace topology is by definition the topology such that a subspace of $X$ is closed iff it's the intersection of $X$ with a closed subspace of $\overline{X}$, and $X$ is the intersection of $X$ with the closed subspace $\overline{X}$.