Germ in math

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เพื่อนเพื่อนได้ รอบรู้ สู้ชีวี

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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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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 have some sense.

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