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# Germ

###### Definition 1 (Germ).

Let $M$ and $N$ be manifolds and $x\in M$. We consider all smooth
mappings $f:U_{f}\to N$, where $U_{f}$ is some open neighborhood of
$x$ in $M$. We define an equivalence relation on the set of mappings
considered, and we put $f\underset{x}{\sim}g$ if there is some
open neighborhood $V$ of $x$ with $f|_{V}=g|_{V}$. The equivalence
class of a mapping $f$ is called the *germ of f at x*, denoted
by $\overline{f}$ or, sometimes, $germ_{x}f$, and we write

$\overline{f}:(M,x)\to(N,f(x))$ |

###### Remark 1.

Germs arise naturally in differential topolgy. It is very convenient
when dealing with derivatives at the point $x$, as every mapping in
a germ will have the same derivative values and properties in $x$,
and hence can be identified for such purposes: every mapping in a
germ gives rise to the same *tangent vector* of $M$ at $x$.

## Mathematics Subject Classification

53B99*no label found*

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