On Bredon's trick: from local to global properties

Abstract: One of the most interesting problems that arise when studying certain structures on topological spaces and in particular on differential manifolds, is to be able to extend the properties that are valid locally to the whole space. A useful tool, which has perhaps been underestimated, is a lemma introduced by G. Bredon, which we refer to as Bredon's trick and which allows the extension of local properties to certain topological spaces. We make a review of this result and show its application in the context of De Rham's cohomology, we will see how this trick allows to give natural alternative demonstrations to classic results, as well as it is fundamental in other cases such as in stratified pseudo-manifolds.