  The idea of a space with smooth structure is a generalization of an idea of a
manifold. K. T. Chen introduced such a space as a differentiable space in his
study of a loop space to employ the idea of iterated path integrals
\cite{Chen:73,Chen:75,Chen:77,Chen:86}. Following the pattern established by
Chen, J. M. Souriau \cite{Souriau:80} introduced his version of a space with
smooth structure, which is called a diffeological space. These notions are
strong enough to include all the topological spaces. However, if one tries to
show de Rham theorem, he must encounter a difficulty to obtain a partition of
unity and thus the Mayer-Vietoris exact sequence in general. In this paper, we
introduce a new version of differential forms to obtain a partition of unity,
the Mayer-Vietoris exact sequence and a version of de Rham theorem in general.
In addition, if we restrict ourselves to consider only CW complexes, we obtain
de Rham theorem for a genuine de Rham complex, and hence the genuine de Rham
cohomology coincides with the ordinary cohomology for a CW complex.
