# relation between germ space and generalized germ space

Let $X$, $Y$ be a topological spaces^{} and $x\in X$. Consider the germ space (http://planetmath.org/GermSpace) and the generalized germ space (http://planetmath.org/GermSpace) at $x$:

$${G}_{x}(X,Y);{G}_{x}^{*}(X,Y).$$ |

If $f:X\to Y$ is a continuous function^{}, then we have an induced element $[f]\in {G}_{x}^{*}(X,Y)$. It can be easily seen, that if $[f]=[g]\in {G}_{x}(X,Y)$, then $[f]=[g]\in {G}_{x}^{*}(X,Y)$. In particular we have a well-defined mapping

$$\tau :{G}_{x}(X,Y)\to {G}_{x}^{*}(X,Y);$$ |

$$\tau ([f])=[f].$$ |

Proposition^{} 1. $\tau $ is injective^{}.

Proof. Indeed, assume that $\tau ([f])=\tau ([g])$ for some $f,g:X\to Y$. Let ${f}^{\prime}:U\to Y$ and ${g}^{\prime}:{U}^{\prime}\to Y$ be a representatives of $\tau ([f])$ and $\tau ([g])$ respectively. It follows, that there exists an open neighbourhood $V\subseteq X$ of $x$ such that

$${f}_{|V}={f}_{|V}^{\prime}={g}_{|V}^{\prime}={g}_{|V}.$$ |

In particular $[f]=[g]$ in ${G}_{x}(X,Y)$, which completes^{} the proof. $\mathrm{\square}$

Proposition 2. If $X$ is a normal space^{} and $Y$ is a normal absolute retract (for example $Y=\mathbb{R}$), then $\tau $ is onto.

Proof. Assume that $[f]\in {G}_{x}^{*}(X,Y)$ for some $f:U\to Y$. Since $X$ is regular^{} (because it is normal) then there exists an open neighbourhood $V\subseteq X$ such that the closure^{} $\overline{V}\subseteq U$. Now since $X$ is normal and $Y$ is a normal absolute retract, then ${f}_{|\overline{V}}$ can be extended to entire $X$ (by the generalized Tietze extension theorem). It is easily seen that any such extension^{} gives the same element in ${G}_{x}(X,Y)$ (and it is independent on the choice of the representative $f$) and if $F:X\to Y$ is an extension of ${f}_{|\overline{V}}$, then

$$\tau ([F])=[f]$$ |

because ${F}_{|V}={f}_{|V}$. This completes the proof. $\mathrm{\square}$

Remark. If in addition $Y$ is a topological ring (for example $Y=\mathbb{R}$), then it can be easily checked that $\tau $ preserves ring structures^{}. In particular if $X$ is normal and $Y=\mathbb{R}$ or $Y=\u2102$, then $\tau $ is an isomorphism^{} of rings. Also it is a good question whether the assumptions^{} in proposition 2 can be weakened.

Title | relation between germ space and generalized germ space |
---|---|

Canonical name | RelationBetweenGermSpaceAndGeneralizedGermSpace |

Date of creation | 2013-03-22 19:18:23 |

Last modified on | 2013-03-22 19:18:23 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 5 |

Author | joking (16130) |

Entry type | Theorem |

Classification | msc 53B99 |