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Concept
Germ (mathematics)
Summary
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 that 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 functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed (the 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 has some meaning.
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 (where Y is any set), then and 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 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 that 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.
Official source
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Ontological neighbourhood