Editing Serre–Swan theorem (section)

== Topology ==
Suppose ''X'' is a compact [[Hausdorff space]], and C(''X'') is the ring of [[continuous function (topology)|continuous]] real-valued functions on ''X''. Analogous to the result above, the category of real vector bundles on ''X'' is equivalent to the category of finitely generated projective modules over C(''X''). The same result holds if one replaces "real-valued" by "complex-valued" and "real vector bundle" by "complex vector bundle", but it does not hold if one replace the field by a [[totally disconnected]] field like the [[rational number]]s.

In detail, let Vec(''X'') be the [[category theory|category]] of [[complex vector bundle]]s over ''X'', and let ProjMod(C(''X'')) be the category of [[finitely generated module|finitely generated]] projective modules over the [[Cstar algebra|C*-algebra]] C(''X''). There is a [[functor]] Γ : Vec(''X'') → ProjMod(C(''X'')) which sends each complex vector bundle ''E'' over ''X'' to the C(''X'')-module Γ(''X'', ''E'') of [[section (fiber bundle)|sections]]. If <math>\tau : (E_1, \pi_1) \to (E_2, \pi_2)</math> is a [[Vector bundle#Vector bundle morphisms|morphism of vector bundles over ''X'']] then <math>\pi_2 \circ \tau = \pi_1</math> and it follows that

:<math>\forall s \in \Gamma(X, E_1) \quad \pi_2 \circ \tau \circ s = \pi_1 \circ s = \text{id}_X,</math>

giving the map 

:<math>\begin{cases} \Gamma \tau : \Gamma(X, E_1) \to \Gamma(X, E_2) \\ s \mapsto \tau \circ s \end{cases}</math>

which respects the module structure {{abbr|(Várilly, 97)|Várilly, Josef C. (1997), ''An introduction to noncommutative geometry'', eprint {{arXiv|physics/9709045v1}} [math-ph]: 6-7. Retrieved 4 June 2015.}}. Swan's theorem asserts that the functor Γ is an [[equivalence of categories]].