errata

The difference is only manifest when the space B is not locally connected. In this case we may have homeomorphisms from X to X over B which do not preserve this structure (Spanier, Algebraic Topology).

is not quite it should be, there is a clear notion of isomorphism of trivialization structure, and a same space X over B may have non isomorphic structures. Alternatively, a continuous function over B does not necessarily preserve the trivialization structures. however, if B is locally connected, trivialization structures are like a pure property of X. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]