  Our main problem is to find finite topological spaces to within
homeomorphism, given (also to within homeomorphism) the quotient-spaces
obtained by identifying one point of the space with each one of the other
points. In a previous version of this paper, our aim was to reconstruct a
topological space from its quotient-spaces; but a reconstruction is not always
possible either in the sense that several non-homeomorphic topological spaces
yield the same quotient-spaces, or in the sense that no topological space
yields an arbitrarily given family of quotient-spaces. In this version of the
paper we present an algorithm that detects, and deals with, all these
situations.
