  In the first half of twentieth century the theory of complex analytic
functions and of their zerosets was fully developed. The definition of
holomorphic function has a local nature. Germs of holomorphic functions form a
distinguished subring of the ring of germs of continuous functions. This way
came out the notion of analytic space. The definition of a complex analytic set
is by local models as in the case of complex manifolds. But while local models
for manifolds are open sets of ${\mathbb C}^n$, a local model of an analytic
space is the zeroset of finitely many analytic functions on an open set of
${\mathbb C}^n$ together with a sheaf of continuous function to be called
holomorphic.
  Towards the years $50$ of the last century, Cartan, Whitney, Bruhat and
others tried to formulate the notion of analytic space over ${\mathbb R}$.
Immediately they realize that the real sets verifying a definition similar to
the complex one form a cathegory whose elements can get unpleasant behaviour.
In particular this cathegory does not get the good properties of complex
analytic spaces, as for instance coherence of their structural sheaves and
Theorems A and B do not hold in general.
