  Paul Erd\H{o}s and L\'aszl\'o Lov\'asz established that any \emph{maximal
intersecting family of $k-$sets} has at most $k^{k}$ blocks. They introduced
the problem of finding the maximum possible number of blocks in such a family.
They also showed that there exists a maximal intersecting family of $k-$sets
with approximately $(e-1)k!$ blocks. Later P\'eter Frankl, Katsuhiro Ota and
Norihide Tokushige used a remarkable construction to prove the existence of a
maximal intersecting family of $k-$sets with at least $(\frac{k}{2})^{k-1}$
blocks. In this article we introduce the notion of a \emph{closed intersecting
family of $k-$sets} and show that such a family can always be embedded in a
maximal intersecting family of $k-$sets. Using this result we present two
examples which disprove two special cases of one of the conjectures of Frankl
et al. This article also provides comparatively simpler construction of maximal
intersecting families of $k-$sets with at least $(\frac{k}{2})^{k-1}$ blocks.
