  The main result of this paper is the proof that all the symmetric products of
a finite Galois-Maximal space are Galois-Maximal spaces. This applies to the
special case of real algebraic varieties, solving the problem first stated by
Biswas and D'Mello in \cite{biswas&d'mello:symmetric_products_M-curves} about
symmetric products of Maximal curves. We also give characterisations of these
spaces and state a new definition that generalises to a larger class of spaces.
Then, we extend the characterisation in terms of the Borel cohomology given in
\cite{us} to the new family. Finally, we introduce the notion of cohomological
stability and cohomological splitting, provide a systematic treatment and
relate them with the properties of being a Maximal or Galois-Maximal space.
These cohomological properties play an important role in the proof of our main
theorem.
