Fibrewise topological spaces

To begin with we work in the category of fibrewise sets over a given set, called the base set. If the base set is denoted by B then a fibrewise set over B consists of a set X together with a function p: X → B, called the projection. For each point b of B the fibre over b is the subset Xb = p-1(b) of X; fibres may be empty since we do not require p to be surjective. Also for each subset B′ of B we regard XB′ = p-1B′ as a fibrewise set over B′ with the projection determined by p. The alternative notation X|B′ is sometimes convenient.

We regard B as a fibrewise set over itself using the identity as projection. Moreover we regard the Cartesian product B × T, for any set T, as a fibrewise set over B using the first projection.

Let X be a fibrewise set over B with projection p. Then X′ is a fibrewise set over B with projection pα for each set X′ and function α:X′ → X; in particular X′ is a fibrewise set over B with projection p|X′ for each subset Xʹ of X. Also X is a fibrewise set over B′ with projection βp for each set B′ and function β:B → B′ in particular X is a fibrewise set over B′ with projection given by p for each superset B′ of B.