# Map germ

Here we define the mathematical concept of a

**map germ**. Given the set of all maps from onemanifold to another we can collect maps together under anequivalence relation . Theseequivalence classes are called map germs.Consider two

manifold s $M$ and $N.$ Let $U\; subseteq\; M$ and $V\; subseteq\; M$ be open neighbourhoods of the point $x\; in\; M.$ Let $f\; :\; U\; o\; N$ and $g\; :\; V\; o\; N.$ We may induce anequivalence relation on the space of mappings $M\; o\; N$ as follows: we say that $f\; sim\; g$ if there exists an open $W\; subseteq\; U\; cap\; V$ such that $f|\_\{W\}\; equiv\; g|\_\{W\},$ i.e. the restriction of $f$ to $W$ coincides with the restriction of $g$ to $W.$The

equivalence class es $[f]$ are called a map germs. The map germ may be denoted by a single representative. If $f(x)\; =\; y$ then we write$f\; :\; (M,x)\; o\; (N,y)$to denote theequivalence class of $f.$**References*** [

*http://mathworld.wolfram.com/MapGerm.html Map Germs at Mathworld*]

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