  We introduce a topological variant of the Grothendieck construction which
serves to represent every fiber bundle over an Alexandroff space. Using this
result we give a classification theorem for fiber bundles over Alexandroff
spaces with T$_0$ fiber and we construct a universal bundle for bundles with
T$_0$ fiber over posets which are cofibrant objects of the category of small
categories. Moreover, we prove that our construction induces an equivalence of
categories between a suitable category of functors and the category of fiber
bundles over a fixed Alexandroff space. In addition, we prove that any fiber
bundle over an Alexandroff space is a fibration.
