Among the subspaces $Y$ , $Z$, and $Y + Z$, which is/are closed in $X$?

Question

Let $X = c_0$ be the space of complex sequences that converge to zero, equipped with the supremum norm. Consider the linear subspaces of $X$:

$Y := \{ \{a_j\}: a_{2j-1} = 0$ for each $j = 1,2,3,...\}$;

$Z := \{ \{b_j\}: b_{2j} = j_2b_{2j-1} = 0$ for each $j = 1,2,3,...\}$.

Among the subspaces $Y$ , $Z$, and $Y + Z$, which is/are closed in $X$?