germ space

Let $X$ , $Y$ be topological spaces and $x\in X$ . Consider the set of all continuous functions

$C(X,Y)=\{f:X\to Y\ |\ f\mbox{ is continuous}\}.$

For any two functions $f,g:X\to Y$ we put

$f\sim_{x}g$

if and only if there exists an open neighbourhood $U\subseteq X$ of $x$ such that

$f_{|U}=g_{|U}.$

The corresponding quotient set is called the germ space at $x\in X$ and we denote it by $G_{x}(X,Y)$ .

More generally, if $X$ , $Y$ are topological spaces with $x\in X$ , then consider the following set:

$C_{x}(X,Y)=\{f:U\to Y\ |\ f\mbox{ is continuous and }U\mbox{ is an open % neighbourhood of }x\}.$

Again we define a relation on $C_{x}(X,Y)$ . If $f:U\to Y$ and $g:U^{\prime}\to Y$ , then put

$f\sim_{x}g$

if and only if there exists and open neighbourhood $V\subseteq X$ of $x$ such that $V\subseteq U\cap U^{\prime}$ and

$f_{|V}=g_{|V}.$

The corresponding set is called the generalized germ space at $x\in X$ and we denote it by $G_{x}^{*}(X,Y)$ .

Note that if $Y=\mathbb{R}$ or $Y=\mathbb{C}$ (or $Y$ is any topological ring), then both $G_{x}(X,Y)$ and $G_{x}^{*}(X,Y)$ have a well-defined ring structure via pointwise addition and multiplication.