On some properties of Musielak-Orlicz sequence spaces

We consider a nontrivial vector space \(X\) and a semimodular \(M\colon X\tp [0, \infty]\) with property: \((\forall\ x \in X) (\exists\ \alpha \gt 0)\ M (\alphax) \lt \infty\) (in other words, \(M\) is normal (i.e. \((\forall\ x\in X \setminus \{0\}) (\exists \alpha \gt 0)\ M (\alphax) \gt 0)\) pregenfunction). The function \(M\) generates in \(X\) a metric \(d\) with \[ d(x, y) := inf \{a \gt 0: M (a^{-1} (x-y)) \leq a\}. \] At the same time \(M\) generates a metric \(\rho\) in Musielak-Orlicz sequence space \(l_M\), namely \[ \rho(\varphi, \psi) := inf \{a \gt 0 : I(a^{-1} (\varphi - \psi)) \leq a\} \] with \(I(\varphi) = \sum_{n \geq 1} M (\varphiφ(n))\). It is proved that the space \((l_M,\rho)\) is complete if and only if the space \((X, d)\) is complete. We consider also the closed subspace \(G_M \subset l_M\) of sequences \(\varphi = \{\varphi(n)\}\) such that \((\forall \alpha \gt 0) (\exists m \in N) \sum_{n\geq m} M(\alpha\varphi(n)) \lt \infty\) and prove that \((G_M ,\rho)\) is separable if and only if \((X, d)\) is the same. Several examples are considered.