  Continuing [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and
Ottenbreit[11]], we further study reflection principles in connection with the
L\"owenheim-Skolem Theorems of stationary logics. In this paper, we mainly
analyze the situations in the models obtained by mixed support iteration of a
supercompact length and then collapsing another supercompact cardinal to make
it $(2^{\aleph_0})^+$. We show, among other things, that the reflection down to
$< 2^{\aleph_0}$ of the non-metrizability of topological spaces with small
character is independent from the reflection properties studied in [Fuchino,
Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]].
