Étale space of continuous functions

Definition

Let $X$ and $Y$ be topological spaces. The étale space of continuous functions at $X$ is a topological space along with an étale map down to $X$, which arises from the sheaf of continuous functions from $X$ to $Y$. Some explicit aspects of this map:

• The fiber of the map at any point $x\in X$, is the set of germs, at $x$, of continuous functions from open neighbourhoods of $X$ to $Y$. In other words, it is the stalk at $x$ for the sheaf of continuous functions.
• The topology on the étale space is given as follows: for every continuous function $f$ from an open subset $U$ of $X$ to $Y$, the set of germs of $f$ at points of $X$ is deemed to be an open subset. Note that this collection of open subsets is closed under taking finite intersections, and arbitrary unions.

Note further that the topology on each fiber is the discrete topology.

Properties

• In most situations, the étale space of continuous functions is not Hausdorff. For full proof, refer: Étale space of continuous functions is not Hausdorff