  A topological space $X$ is called almost discretely Lindel\"of if every
discrete set $D \subset X$ is included in a Lindel\"of subspace of $X$. We say
that the space $X$ is {\em $\mu$-sequential} if for every non-closed set $A
\subset X$ there is a sequence of length $\le \mu$ in $A$ that converges to a
point which is not in $A$. With the help of a technical theorem that involves
elementary submodels, we establish the following two results concerning such
spaces.
  (1) For every almost discretely Lindel\"of $T_3$ space $X$ we have $|X| \le
2^{\chi(X)}$.
  (2) If $X$ is a $\mu$-sequential $T_2$ space of pseudocharacter $\psi(X) \le
2^\mu$ and for every free set $D \subset X$ we have $L(\overline{D}) \le \mu$,
then $|X| \le 2^\mu$.
  The case $\chi(X) = \omega$ of (1) provides a solution to Problem 4.5 from
"I. Juh\'asz, V. Tkachuk, and R. Wilson, Weakly linearly Lindel\"of
monotonically normal spaces are Lindel\"of", while the case $\mu = \omega$ of
(2) is a partial improvement on the main result of "A.V. Archangel'skii and
R.Z. Buzyakova, On some properties of linearly Lindel\"of spaces".
