  Among plenty of applications, low-dimensional homogeneous spaces appear in
cosmological models as both, classical factor spaces of multidimensional
geometry and minisuperspaces in canonical quantization.
  Here a new tool to restrict their continuous deformations is presented:
Classifying spaces for homogeneous manifolds and their related Lie isometry
deformations.
  The adjoint representation of n-dimensional real Lie algebras induces a
natural topology on their classifying space, which encodes the natural
algebraic relationship between different Lie algebras therein. For n>1 this
topology is not Hausdorffian. Even more it satisfies only the separation axiom
T_0, but not T_1, i.e. there is a constant sequence which has a limit different
from the members of the sequence. Such a limit is called a transition.
  Recently it was found that transitions are the natural generalization and
transitive completion of the well-known In\"on\"u-Wigner contractions. For n<5
the relational classifying spaces are constructed explicitly.
  Calculating their characteristic scalar invariants via triad representations
of the characteristic isometry, local homogeneous Riemannian 3-spaces are
classified in their natural geometrical relations to each other. Their
classifying space is a composition of pieces with different isometry types.
Although it is Hausdorffian, different topological transitions to the same
limit may induce locally non-Euclidean regions (e.g. at Bianchi tppes VII_0).
