  A space $X$ is said to be "cellular-Lindel\"of" if for every cellular family
$\mathcal{U}$ there is a Lindel\"of subspace $L$ of $X$ which meets every
element of $\mathcal{U}$. Cellular-Lindel\"of spaces generalize both Lindel\"of
spaces and spaces with the countable chain condition. Solving questions of Xuan
and Song, we prove that every cellular-Lindel\"of monotonically normal space is
Lindel\"of and that every cellular-Lindel\"of space with a regular
$G_\delta$-diagonal has cardinality at most $2^\mathfrak{c}$. We also prove
that every normal cellular-Lindel\"of first-countable space has cardinality at
most continuum under $2^{<\mathfrak{c}}=\mathfrak{c}$ and that every normal
cellular Lindel\"of space with a $G_\delta$-diagonal of rank $2$ has
cardinality at most continuum.
