germ space

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

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

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:

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

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 $Y=ℝ$ or $Y=ℂ$ (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.