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∈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.
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