Nilpotent space
In topology, a branch of mathematics, a nilpotent space, first defined by Emmanuel Dror (1969), is a based topological space "X" such that Simply connected spaces and simple spaces are (trivial) examples of nilpotent spaces; other examples are connected loop spaces. The homotopy fiber of any map between nilpotent spaces is a disjoint union of nilpotent spaces. Moreover, the null component of the pointed mapping space formula_8, where "K" is a pointed, finite-dimensional CW complex and "X" is any pointed space, is a nilpotent space. The odd-dimensional real projective spaces are nilpotent spaces, while the projective  plane is not. A basic theorem about nilpotent spaces states that any map that induces an integral homology isomorphism between two nilpotent space is a weak homotopy equivalence. For simply connected spaces, this theorem recovers a well-known corollary to the Whitehead and Hurewicz theorems. Nilpotent spaces are of great interest in rational homotopy theory, because most constructions applicable to simply connected spaces can be extended to nilpotent spaces. The Bousfield–Kan nilpotent completion of a space associates with any connected pointed space "X" a universal space formula_9 through which any map of "X" to a nilpotent space "N" factors  uniquely up to a contractible space of choices. Often, however, formula_9 itself is not nilpotent but only an inverse limit of a tower of nilpotent spaces. This tower, as a pro-space, always models the homology type of the given pointed space "X". Nilpotent spaces admit a good arithmetic localization theory in the sense of Bousfield and Kan cited above, and the unstable Adams spectral sequence strongly converges for any such space. Let "X" be a nilpotent space and let "h" be a reduced generalized homology theory, such as K-theory. If "h"("X")=0, then "h" vanishes on any Postnikov section  of "X". This follows from a theorem that states that any such section is "X"-cellular.