We briefly recall the construction of the étalé space of a sheaf. Given a sheaf over a space , the associated étalé space is a topological space which, as a set, is the disjoint union of the stalks of , with the topology induced by the basis , where is an open set in , and is the germ at of the element .
Given a topological space , consider the sheaf of real-valued continuous functions on , which we denote . It is more or less easy to see that the associated étalé space is not Hausdorff iff the space has the following property :
There exist continuous functions , where is an open set, and a point such that the germs of and at are different, but nevertheless for any open neighborhood of there exists an open set such that .
For instance, has property , and it’s easy to see that is inherited by subspaces and products which contain any factor. This, for instance, shows that subspaces of are spaces satisfying this property.
Is there any characterization of all topological spaces for which the étalé space associated with the sheaf of real-valued continuous functions is not Hausdorff?