Vector bundles and projective modules

Serre and Swan showed that the category of vector bundles over a compact space X is equivalent to the category of finitely generated projective modules over the ring of continuous functions on X. In this paper, titled after the famous paper by Swan, this result is extended to an arbitrary topological space X. Also the well-known homotopy classification of the vector bundles over compact X up to isomorphism is extended to arbitrary X. It is shown that the Ko-functor and the Witt group of the ring of continuous functions on X coincide, and they are homotopy-type invariants of X.