Map germ

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Here we define the mathematical concept of a map germ. Given the set of all maps from one manifold to another we can collect maps together under an equivalence relation. These equivalence classes are called map germs.

Consider two manifolds $M$ and $N .$ Let $U \subseteq M$ and $V \subseteq M$ be open neighbourhoods of the point $x \in M.$ Let $f : U \to N$ and $g : V \to N.$ We may induce an equivalence relation on the space of mappings $M \to N$ 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 classes $[f]$ 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) \to (N,y) \$ to denote the equivalence class of $f .$