Germs of smooth functions

In maths, the notion of germ is used to definition “generic” of functions, in other hand, germ is used to say about mostly functions on topological spaces.

Consider to a point, we need to get information from that point, so we take a neighborhood of that point, assume that the point is x.

With a smooth function on manifold M, germ of smooth functions near x is a pair (U,f) such that f: U \to \mathbb{R} is smooth on local neighborhood U of x.

Two pairs (U,f) and (V,g) is equivalent if there is an open subset W \subset U \cap V such that f_{|_W} = g_{|_W}. So a germ of smooth function near x is an equivalent class of that pairs.

(to be cont.)

References: Wikipedia,