Separating the étalé space

We briefly recall the construction of the étalé space of a sheaf. Given a sheaf F over a space X, the associated étalé space is a topological space E which, as a set, is the disjoint union of the stalks of F, with the topology induced by the basis B(s, U) = \{s_x : x\in U\}, where U is an open set in X, s\in F(U) and s_x is the germ at x of the element s.

Given a topological space X, consider the sheaf of real-valued continuous functions on X, which we denote C. It is more or less easy to see that the associated étalé space is not Hausdorff iff the space X has the following property (\star):

There exist continuous functions f,g:U\to\mathbb{R}, where U is an open set, and a point x\in U such that the germs of f and g at x are different, but nevertheless for any open neighborhood V\subseteq U of x there exists an open set W\subseteq V such that f|_W = g|_W.

For instance, \mathbb{R} has property (\star), and it’s easy to see that (\star) is inherited by subspaces and products which contain any (\star) factor. This, for instance, shows that subspaces of \mathbb{R}^n 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?

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Separating the étalé space

Self isometries of a compact metric space

Consider an isometry from a compact metric space to itself, this is simply a function that preserves distances. Notice that this implies that the function is continuous and moreover that it is injective. What we have to prove is that such a function is also surjective. Observe that this implies that the function is indeed a homeomorphism.

A consequence of this fact is that the category that has compact metric spaces as objects and isometries as arrows enjoys a Cantor-Schröder-Bernstein-like property.

Self isometries of a compact metric space