Germ (mathematics)

In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind which captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets. In specific implementations of this idea, the sets or maps in question will have some property, such as being analytic or smooth, but in general this is not needed (the maps or functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word local have some sense.

The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain.

Given a point x of a topological space X, and two maps f, g : X → Y (where Y is any set), then f and g define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal; meaning that f(u) = g(u) for all u in U. Similarly, if S and T are any two subsets of X, then they define the same germ at x if there is again a neighbourhood U of x such that

It is straightforward to see that defining the same germ at x is an equivalence relation (be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written

Given a map f on X, then its germ at x is usually denoted [f ]_x. Similarly, the germ at x of a set S is written [S]_x. Thus,

A map germ at x in X which maps the point x in X to the point y in Y is denoted as

When using this notation, f is then intended as an entire equivalence class of maps, using the same letter f for any representative map.

Notice that two sets are germ-equivalent at x if and only if their characteristic functions are germ-equivalent at x:

Maps need not be defined on all of X, and in particular they don't need to have the same domain. However, if f has domain S and g has domain T, both subsets of X, then f and g are germ equivalent at x in X if first S and T are germ equivalent at x, say $\scriptstyle S\,\cap \,U\;=\;T\,\cap \,U$ $\scriptstyle S\,\cap\,U\;=\;T\,\cap\,U$ , and then moreover $\scriptstyle f|_{S\cap V}=g|_{T\cap V}$ $\scriptstyle f|_{S\cap V} = g|_{T\cap V}$ , for some smaller neighbourhood V with $\scriptstyle x\in V\subset U$ $\scriptstyle x\in V \subset U$ . This is particularly relevant in two settings:

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