Topological criterium for characterizing some spaces

Here we give a characterization for topological properties of subspaces of the product space R-N of sequences of reals. For analytic ideals, i.e. analytic vector subspaces X subset of or equal to R-N which verify: [x is an element of X and For All n, \y(n)\ less than or equal to \x(n)\] double right arrow y is an element of X, we have the following dichotomy: X admits a polish vector space topology stronger than the product topology of R-N or one can embed into X in a strong sense the space of finite sequences c(00) or the space l(infinity). If X admits such a Polish topology we find special functions which define this topology and have a simple form; we name them "evaluations functions" and with this notion we specify the descriptive complexity of X and we give some properties of ideals which only admit a complete metrizable topology. (C) 1999 Academie des Sciences/Editions scientifiques et medicales Elsevier SAS.