relation between germ space and generalized germ space

Let X, Y be a topological spacesMathworldPlanetmath and xX. Consider the germ space ( and the generalized germ space ( at x:


If f:XY is a continuous functionMathworldPlanetmathPlanetmath, then we have an induced element [f]Gx*(X,Y). It can be easily seen, that if [f]=[g]Gx(X,Y), then [f]=[g]Gx*(X,Y). In particular we have a well-defined mapping


Proof. Indeed, assume that τ([f])=τ([g]) for some f,g:XY. Let f:UY and g:UY be a representatives of τ([f]) and τ([g]) respectively. It follows, that there exists an open neighbourhood VX of x such that


In particular [f]=[g] in Gx(X,Y), which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Proposition 2. If X is a normal spaceMathworldPlanetmath and Y is a normal absolute retract (for example Y=), then τ is onto.

Proof. Assume that [f]Gx*(X,Y) for some f:UY. Since X is regularPlanetmathPlanetmathPlanetmathPlanetmath (because it is normal) then there exists an open neighbourhood VX such that the closureMathworldPlanetmathPlanetmath V¯U. Now since X is normal and Y is a normal absolute retract, then f|V¯ can be extended to entire X (by the generalized Tietze extension theorem). It is easily seen that any such extensionPlanetmathPlanetmath gives the same element in Gx(X,Y) (and it is independent on the choice of the representative f) and if F:XY is an extension of f|V¯, then


because F|V=f|V. This completes the proof.

Remark. If in addition Y is a topological ring (for example Y=), then it can be easily checked that τ preserves ring structuresMathworldPlanetmath. In particular if X is normal and Y= or Y=, then τ is an isomorphismPlanetmathPlanetmathPlanetmathPlanetmath of rings. Also it is a good question whether the assumptionsPlanetmathPlanetmath 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