The envelope of this family of lines is a plane curve whose inverse image under the development is the edge of regression.
22.
They also demonstrate contravariant behavior with respect to proper maps-that is, maps such that the inverse image of every compact set is compact.
23.
In such cases the inverse image operation is often compatible with composition of these maps between objects, or in more technical terms is a functor.
24.
In order to remedy this, one defines in this situation for a sheaf of \ mathcal O _ Y-modules \ mathcal G its inverse image by
25.
It's instructive to do it also with the general topological definition of continuous functions ( inverse images of open sets are open ), where sequences are unnecessary.
26.
A function, then the quotient topology on " Y " is the collection of subsets of " Y " that have open inverse images under " f ".
27.
This introduction of some " slack " in the system of inverse images causes some delicate issues to appear, and it is this set-up that fibred categories formalise.
28.
As part of the predominating Grothendieck approach, there are three corresponding notions of " fiber " of a morphism : the first being the simple inverse image of a point.
29.
A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers.
30.
Note that each set in the countable intersection is an element of ? because it is the inverse image of a Borel subset under a ?-measurable function f _ k.
