  We use the Leray spectral sequence for the sheaf cohomology groups with
compact supports to obtain a vanishing result. The stalks of sheaves
$R^{\bullet}\phi_{!}\mathcal{O}$ for the structure sheaf $\mathcal{O}$ on the
total space of a holomorphic fiber bundle $\phi$ has canonical topology
structures. Using the standard \vCech argument we prove a density lemma for
QDFS-topology on this stalks. In particular, we obtain a vanishing result for
holomorphic fiber bundles with Stein fibers. Using K\"unnet formulas,
properties of an inductive topology (with respect to the pair of spaces) on the
stalks of the sheaf $R^{1}\phi_{!}\mathcal{O}$ and a cohomological criterion
for the Hartogs phenomenon we obtain the main result on the Hartogs phenomenon
for the total space of holomorphic fiber bundles with (1,0)-compactifiable
fibers.
