In Grauert & Remmert - "Coherent Analytic Sheaves" it is stated on page 34 that "every point of an arbitrary complex space has a neighborhood basis of STEIN spaces". Since I am not familiar with complex spaces or stein spaces and stein manifolds, I was wondering if this is true for complex manifolds as well, i.e.
Does every point of a complex manifold have a neighborhood basis of stein manifolds?
Let $M$ be a complex $n$-dimensional manifold and $p \in M$. There is an open neighbourhood $U$ of $p$ and an open set $V \subseteq \mathbb{C}^n$ containing $0$ and a biholomorphism $\varphi : U \to V$ with $\varphi(p) = 0$. Let $d(0, \partial V) = D$, which is necessarily positive. Note that for $d_1, \dots, d_n$ satisfying $d_1^2 + \dots + d_n^2 < D^2$, the polydisc $D_{d_1, \dots, d_n} = \{z \in \mathbb{C}^n \mid |z_1| < d_1, \dots, |z_n| < d_n\}$ is contained in $V$. So $\varphi^{-1}(D_{d_1,\dots,d_n})$ is an open neighbourhood of $p$ in $M$ biholomorphic to a polydisc. As polydiscs are Stein, we're done.