  We prove, via transfinite recursion, the existence, inside any linearly
ordered set of appropriate regular cardinality $\lambda$, of a particular kind
of well-ordered subsets characterized by the property of $\lambda$-fullness.
Let $H$ be a set of regular cardinals: by using our results about well-ordered
$\lambda$-full sets we show that if $\inf H$ is a weakly compact cardinal,
then, for every LOTS $X$, $H$-compactness is equivalent to the nonexistence of
gaps of types in $H$.
