germ space

Let X, Y be topological spacesMathworldPlanetmath and xX. Consider the set of all continuous functionsMathworldPlanetmathPlanetmath

C(X,Y)={f:XY|f is continuous}.

For any two functions f,g:XY we put


if and only if there exists an open neighbourhood UX of x such that


The corresponding quotient set is called the germ space at xX and we denote it by Gx(X,Y).

More generally, if X, Y are topological spaces with xX, then consider the following set:

Cx(X,Y)={f:UY|f is continuous and U is an open neighbourhood of x}.

Again we define a relation on Cx(X,Y). If f:UY and g:UY, then put


if and only if there exists and open neighbourhood VX of x such that VUU and


The corresponding set is called the generalized germ space at xX and we denote it by Gx*(X,Y).

Note that if Y= or Y= (or Y is any topological ring), then both Gx(X,Y) and Gx*(X,Y) have a well-defined ring structureMathworldPlanetmath via pointwise addition and multiplicationPlanetmathPlanetmath.

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