  We introduce smooth L^\infty differential forms on a singular (semialgebraic)
set X in R^n. Roughly speaking, a smooth L^\infty differential form is a
certain class of equivalence of 'stratified forms', that is, a collection of
smooth forms on disjoint smooth subsets (stratification) of X with matching
tangential components on the adjacent strata and bounded size (in the metric
induced from R^n). We identify the singular homology of X as the homology of
the chain complex generated by semialgebraic singular simplices, i.e.
continuous semialgebraic maps from the standard simplices into X. Singular
cohomology of X is defined as the homology of the Hom dual to the chain complex
of the singular chains. Finally, we prove a De Rham type theorem establishing a
natural isomorphism between the singular cohomology and the cohomology of
smooth L^\infty forms.
