Banach bundle (non-commutative geometry)

In mathematics, a Banach bundle is a fiber bundle over a topological Hausdorff space, such that each fiber has the structure of a Banach space.
Definition.
Let formula_1 be a topological Hausdorff space, a (continuous) Banach bundle over formula_2 is a tuple formula_3, where formula_4 is a topological Hausdorff space, and formula_5 is a continuous, open surjection, such that each fiber formula_6 is a Banach space. Which satisfies the following conditions:
If the map formula_20 is only upper semi-continuous, formula_21 is called upper semi-continuous bundle.
Examples.
Trivial bundle.
Let "A" be a Banach space, "X" be a topological Hausdorff space. Define formula_22 and formula_5 by formula_24. Then formula_25 is a Banach bundle, called the trivial bundle