  In this paper, we introduce the notions of $\alpha$-quasicomplemented and
totally $\alpha$-quasicomplemented subspaces and we established some results
under these contexts. We show, for example, that if $X$ is a separable or
reflexive Banach space and $Y$ is a closed infinite codimensional subspace of
$X$, then $Y$ is totally$\mathit{\ }\alpha$-quasicomplemented if, and only if,
$\alpha<\aleph_{0}$ $\left( \text{this is an analogue of the theorem of
Murray-Mackey and Lindenstrauss}\right) $. We also show that if $H$ is a
Hilbert space and $Y,W\subset H$ are closed subspaces of $H$ such that $W$ is
orthogonal to $Y$ and $\operatorname{codim}\left( Y+W\right) =\infty$, then $Y$
has a quasicomplement $Z$ containing $W$ with $\dim Z/W=\infty$. Other results
in the different contexts are also included. Such results establish a
connection between the theory of quasicomplemented subspaces and $\left(
\alpha,\beta\right) $-spaceability.
