# 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

 $\tau:G_{x}(X,Y)\to G_{x}^{*}(X,Y);$
 $\tau([f])=[f].$

$\tau$ 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

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

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

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. $\square$

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 $Y=\mathbb{C}$, 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 RelationBetweenGermSpaceAndGeneralizedGermSpace 2013-03-22 19:18:23 2013-03-22 19:18:23 joking (16130) joking (16130) 5 joking (16130) Theorem msc 53B99