  A family of closed subsets of a topological space $X$ is called a (strict)
$Cld$-fan in $X$ if this family is (strictly) compact-finite but not locally
finite in $X$. Applications of (strict) $Cld$-fans are based on a simple
observation that $k$-spaces contain no $Cld$-fan and Ascoli spaces contain no
strict $Cld$-fan. In this paper we develop the machinery of (strict) fans and
apply it to detecting the $k$-space and Ascoli properties in spaces that
naturally appear in General Topology, Functional Analysis, and Topological
Algebra. In particular, we detect (generalized) metric spaces $X$ whose
functor-spaces, functions spaces, free (para)topological (abelian) groups, free
(locally convex) linear topological spaces, free (Lawson) topological
semilattices, and free (para)topological (Clifford, Abelian) inverse semigroups
are $k$-spaces or Ascoli spaces.
