The definition of first selection principle is well known: $S_1(A,B)$. Let $A$ and $B$ be families of sets. The symbol $S_1(A, B)$ denotes the statement: for each sequence $(A_n : n \in \omega)$ of elements of $A$ there is a sequence $(x_n : n\in \omega)$ such that each $x_n$ is an element of $A_n$, and $\{x_n : n\in \omega\}$ is an element of $B$.

Can we assume that $x_{n_k}=\emptyset$ for $k\in \omega$ ?

elements$x_n$ to be empty, which seems "ill typed" (a priori theyarejust elements, not sets). Also, you have only defined $x_n$ in your statement; what is $x_{n_k}$? $\endgroup$