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. We now treat these open subsets as a subbasis for the topology of the étale space.