# 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 $Y=\mathbb{R}$ or $Y=\u2102$ (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^{}.

Title | germ space |
---|---|

Canonical name | GermSpace |

Date of creation | 2013-03-22 19:18:20 |

Last modified on | 2013-03-22 19:18:20 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 53B99 |