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.
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
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.
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.
This is particularly relevant in two settings: If f and g are germ equivalent at x, then they share all local properties, such as continuity, differentiability etc., so it makes sense to talk about a differentiable or analytic germ, etc.
Similarly for subsets: if one representative of a germ is an analytic set then so are all representatives, at least on some neighbourhood of x. Algebraic structures on the target Y are inherited by the set of germs with values in Y.
In the same way, if Y is an abelian group, vector space, or ring, then so is the set of germs.
The set of germs at x of maps from X to Y does not have a useful topology, except for the discrete one.
It therefore makes little or no sense to talk of a convergent sequence of germs.
(finite order Taylor series at x of map(-germs)) do have topologies as they can be identified with finite-dimensional vector spaces.
The idea of germs is behind the definition of sheaves and presheaves.
Typical examples of abelian groups here are: real-valued functions on U, differential forms on U, vector fields on U, holomorphic functions on U (when X is a complex manifold), constant functions on U and differential operators on U.
The reason is that formation of stalks preserves finite limits.
have additional structure, it is possible to define subsets of the set of all maps from X to Y or more generally sub-presheaves of a given presheaf
As a consequence, germs, constituting stalks of sheaves of various kind of functions, borrow this scheme of notation: For germs of sets and varieties, the notation is not so well established: some notations found in literature include: The key word in the applications of germs is locality: all local properties of a function at a point can be studied by analyzing its germ.
They are a generalization of Taylor series, and indeed the Taylor series of a germ (of a differentiable function) is defined: you only need local information to compute derivatives.
Germs are useful in determining the properties of dynamical systems near chosen points of their phase space: they are one of the main tools in singularity theory and catastrophe theory.
When the topological spaces considered are Riemann surfaces or more generally complex analytic varieties, germs of holomorphic functions on them can be viewed as power series, and thus the set of germs can be considered to be the analytic continuation of an analytic function.
Germs can also be used in the definition of tangent vectors in differential geometry.
A tangent vector can be viewed as a point-derivation on the algebra of germs at that point.
[1] As noted earlier, sets of germs may have algebraic structures such as being rings.
The types of local rings that arise, however, depend closely on the theory under consideration.
To see why, observe that the maximal ideal m of this ring consists of all germs that vanish at the origin, and the power mk consists of those germs whose first k − 1 derivatives vanish.
If this ring were Noetherian, then the Krull intersection theorem would imply that a smooth function whose Taylor series vanished would be the zero function.
But this is false, as can be seen by considering This ring is also not a unique factorization domain.
of germs at the origin of continuous functions on R even has the property that its maximal ideal m satisfies m2 = m. Any germ f ∈ m can be written as where sgn is the sign function.
Since |f| vanishes at the origin, this expresses f as the product of two functions in m, whence the conclusion.