an algebro-geometric variant of this construction is applied to the category of algebraic varieties; it associates with a given algebraic variety "x" the grothendieck's k-group of the category of locally free sheaves (or coherent sheaves) on "x". given a compact topological space "x", the topological k-theory k("x") of (real) vector bundles over "x" coincides with "k" of the ring of continuous real-valued functions on "x".