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

Holomorphic extension in holomorphic fiber bundles with (1,0)-compactifiable fiber