We examine the multiplicity of complementation amongst subspaces of $V_\infty$. A subspace $V$ is a complement of a subspace $W$ if $V \cap W = \{0\}$ and $(V \cup W)^\ast = V_\infty$. A subspace is called fully co-r.e. if it is generated by a co-r.e. subset of a recursive basis of $V_\infty$. We observe that every r.e. subspace has a fully co-r.e. complement. Theorem. If $S$ is any fully co-r.e. subspace then $S$ has a decidable complement. We give an analysis of other types of complements $S$ may have. For example, if $S$ is fully co-r.e. and nonrecursive, then $S$ has a (nonrecursive) r.e. nowhere simple complement. We impose the condition of immunity upon our subspaces. Theorem. Suppose $V$ is fully co-r.e. Then $V$ is immune iff there exist $M_1, M_2 \in L(V_\infty)$, with $M_1$ supermaximal and $M_2 k$-thin, such that $M_1 \oplus V = M_2 \oplus V = V_\infty$. Corollary. Suppose $V$ is any r.e. subspace with a fully co-r.e. immune complement $W (e.g., V$ is maximal or $V$ is $h$-immune). Then there exist an r.e. supermaximal subspace $M$ and a decidable subspace $D$ such that $V \oplus W = M \oplus W = D \oplus W = V_\infty$. We indicate how one may obtain many further results of this type. Finally we examine a generalization of the concepts of immunity and soundness. A subspace $V$ of $V_\infty$ is nowhere sound if (i) for all $Q \in L(V_\infty)$ if $Q \supset V$ then $Q = V_\infty$, (ii) $V$ is immune and (iii) every complement of $V$ is immune. We analyse the existence (and ramifications of the existence) of nowhere sound spaces.