Topologically, compact toric varieties can be constructed as identification
spaces: they are quotients of the product of a compact torus and the order
complex of the fan. We give a detailed proof of this fact, extend it to the
non-compact case and draw several, mostly cohomological conclusions.

In particular, we show that the equivariant integral cohomology of a toric
variety can be described in terms of piecewise polynomials on the fan if the
ordinary integral cohomology is concentrated in even degrees. This generalises
a result of Bahri-Franz-Ray. We also investigate torsion phenomena in integral
cohomology.