  In answer to a question on Mathoverflow we show that the Boolean algebra
$\mathcal{P}(\omega)/\mathit{fin}$ contains a family
$\{\mathcal{B}_X:X\subseteq\mathfrak{c}\}$ of subalgebras with the property
that $X\subseteq Y$ implies $\mathcal{B}_Y$ is a subalgebra of $\mathcal{B}_X$
and if $X\not\subseteq Y$ then $\mathcal{B}_Y$ is not embeddable
into~$\mathcal{B}_X$. The proof proceeds by Stone duality and the construction
of a suitable family of separable zero-dimensional compact spaces.
