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 $Y=ℝ$), 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 $Y=ℝ$), then it can be easily checked that $\tau$ preserves ring structures. In particular if $X$ is normal and $Y=ℝ$ or $Y=ℂ$, 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
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