# Germ

2010 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A term signifying a "pointwise localization" of various mathematical objects (germs of functions, germs of mappings, germs of analytic sets, etc.). Let, for example, $x$ be a point in a topological space and let $F$ be some family of functions defined in a neighbourhood of $x$ (each in its own neighbourhood). Two functions $f,g\in F$ are said to be equivalent (at $x$) if they coincide in some neighbourhood of $x$. An equivalence class generated by this relation is called a germ of functions of class $F$ at $x$. In this way are defined the germs of continuous functions, of differentiable functions at the points of a differentiable manifold, of holomorphic functions at the points of a complex manifold, etc. If the family $F$ has some algebraic structure, then the set of germs of functions of the family $F$ inherits this structure (the operations are carried out on representatives of classes). In particular, the germs of holomorphic functions at a point $z$ form a ring. Elements of the quotient field of this ring are called germs of meromorphic functions at $z$.

Similarly one can define a germ of a family of subsets of a topological space. For instance, at the points of an analytic manifold there are germs of analytic sets (the equivalence class is defined by coincidence in a neighbourhood of a given point). On germs of families of subsets set-theoretic operations and relations are naturally defined. The notion of a germ is also meaningful in the case of other objects defined on open subsets of a topological space.

See also Analytic function; Meromorphic function; Sheaf.