Editing Germ (mathematics) (section)

===Basic properties===
If ''f'' and ''g'' are germ equivalent at ''x'', then they share all local properties, such as continuity, [[differentiability]] etc., so it makes sense to talk about a ''differentiable or analytic germ'', etc. Similarly for subsets: if one representative of a germ is an [[analytic set]] then so are all representatives, at least on some neighbourhood of ''x''.

Algebraic structures on the target ''Y'' are inherited by the set of germs with values in ''Y''.  For instance, if the target ''Y'' is a [[Group (mathematics)|group]], then it makes sense to multiply germs: to define [''f'']<sub>''x''</sub>[''g'']<sub>''x''</sub>, first take representatives ''f'' and ''g'', defined on neighbourhoods ''U'' and ''V'' respectively, and define [''f'']<sub>''x''</sub>[''g'']<sub>''x''</sub> to be the germ at ''x'' of the pointwise product map ''fg'' (which is defined on <math>U\cap V</math>).  In the same way, if ''Y'' is an [[abelian group]], [[vector space]], or [[Ring (mathematics)|ring]], then so is the set of germs.

The set of germs at ''x'' of maps from ''X'' to ''Y'' does not have a useful [[Topological space|topology]], except for the [[Discrete topology|discrete]] one. It therefore makes little or no sense to talk of a convergent sequence of germs. However, if ''X'' and ''Y'' are [[manifold]]s, then the spaces of [[Jet (mathematics)|jets]] <math>J_x^k(X,Y)</math> (finite order [[Taylor series]] at ''x'' of map(-germs)) do have topologies as they can be identified with [[finite-dimensional vector space]]s.