Étale space of continuous functions is not necessarily Hausdorff

Statement

Let $X$ and $Y$ be topological spaces. Then the étale space of continuous functions from $X$ to $Y$ need not be Hausdorff, even if $X$ and $Y$ are Hausdorff.

Positive statement

Suppose $X$ is a Hausdorff space. Then, the étale space of continuous functions from $X$ to $Y$ is Hausdorff, if and only if, for any functions $f_{1}:U_{1}\to Y,f_{2}:U_{2}\to Y$ from open subsets of $X$ to $Y$, the set of points in $U_{1}\cap U_{2}$ at which the germs of $f_{1}$ and $f_{2}$ are equal, is a closed subset of $U_{1}\cap U_{2}$.

Contrapositive statement

This form of the statement is more useful from the point of view of constructing counterexamples:

Suppose we can find two continuous maps $f,g:U\to Y$ where $U$ is an open subset of $X$ containing a specific point $x\in X$, such that $f$ and $g$ do not have the same germ at $x$, but such that for every open set $V\subset U$ containing $x$, there exists $x'\in V$ such that $f,g$ have the same germ at $V$. Then, the étale space of continuous functions from $X$ to $Y$ is not Hausdorff.

Proof

We shall prove the contrapositive version.

Suppose the above condition holds. Then, let $f_{x}$ and $g_{x}$ denote the germs of $f$ and $g$ at $x$. Suppose there are open subsets separating these. Then, by the definition of the topology on the étale space of continuous functions, there exist open subsets $V_{1}$ and $V_{2}$ about $x$ and functions $f',g'$ defined on $V_{1}$ and $V_{2}$, such that:

• $f'_{x}=f_{x}$ and $g'_{x}=g_{x}$
• For any point $x'\in V_{1}\cap V_{2}$, $f'_{x'}\neq g'_{x'}$

Now, by definition of germ, we can find a small open subset $W\subset U\cap V_{1}\cap V_{2}$ about $x$ such that $f=f'$ and $g=g'$ on $W$. But $f'$ and $g'$ have different germs at all points in $W$, hence so do $f$ and $g$. This contradicts our assumption that for every neighbourhood of $x$ there exists $x'$ in the neighbourhood where the germ of $f$ and $g$ are equal.

Explanation

Since $X$ is Hausdorff, it is certainly true that given any two points in different fibers of the étale space, we can separate them by disjoint open subsets. The problem is if we pick two points in the same fiber, such that they are germs of different functions, but the functions for which they are germs, look similar enough at points arbitrarily close to the given point.

In our case, the two points are in the fiber above $x$, and are germs of $f,g$ respectively, with both $f$ and $g$ resembling each other at points arbitrarily close to $x$.

Example

We show that the étale space of continuous functions from $\mathbb {R}$ to $\mathbb {R}$ is not Hausdorff. For the proof, we take $x=0$, $f$ to be the zero function, and $g$ to be the function:

$g(x)=x^{+}=\max\{x,0\}$

Clearly, $f$ and $g$ do not have the same germ at $0$, but they have the same germ at any $x'<0$. Thus, any open neighbourhood of $0$ contains a point where the germ of $f$ equals the germ of $g$.

The same idea can be used for the étale space of continuous functions from any manifold to $\mathbb {R}$.

Similar proofs show that the étale space of smooth maps from a differential manifold to $\mathbb {R}$ is not Hausdorff.