at points of the orbit $G\cdot x$ form the conjugacy class $(G_x)$ of $G_x$ in $G$, which identifies the [*orbit–type*]{} of (the points of) $G\cdot
x$. The set of points $x\in\real^n$ (of $G$-orbits) with the same orbit–type forms an [*isotropy–type stratum of*]{} $(G,\real^
n)$.

The [*orbit space*]{} of the action of $G$ in $\real^n$ is defined as the quotient space $\real^ n/G$, endowed with the quotient topology and differentiable structure. The image of a stratum of $\real^{n}$, through the canonical projection $\pi:\
\real^n\rightarrow\real^n/G $, defines an [*isotropy–type stratum*]{} of $\real^n/G$; the connected components of an [*isotropy–type stratum*]{} of $\real^n/G$ are iso–dimensional manifolds ([*primary strata*]{} of $\real^n/G$), but the orbit space is not a manifold.

Almost all the points of $\real^n/G$ belong to a unique stratum, the [*principal stratum*]{}, which is a connected open dense subset of $\real^ n/G$. The boundary of the principal stratum is the union of disjoint [*singular*]{} strata. Every stratum $\widehat S$ of $\real^n/G$ is open in its topological closure $\overline{\widehat S}$.

The following partial ordering can be introduced in the set of all the orbit–types: $(H)<(K)$, if $H$ is conjugate with a subgroup of $K$. If $(H_i)$ is the orbit–type of the stratum $\widehat
S_i$ of 