On the theory of $q$-complete spaces

In this article, we show that if $X$ is a Stein space of dimension $n$ and, if $D$ is a locally $q$-complete open set in $X$, then $D$ is $q$-complete. This gives, in particular, a positive answer to the local Steinness problem, which is one of the most classical problem in complex analytic geometry, namely, if $D$ is a locally Stein open set in a Stein space $X$, then $D$ is Stein