Does the bundle of germs of functions $f:X\to \mathbb R$ have the same sheaf of sections as $X\times \mathbb R$?

Question

I'm just starting to learn about sheaves, and I'm confused about a certain matter:

I've just learned, to my delight, that every sheaf $S$ on a space $X$ is the sheaf of sections of a particular bundle (by bundle I mean a surjective map $E\to X$), specifically the bundle of stalks of $S$.

So, for example, the sheaf of continuous functions on $X\to \mathbb R$ can be interpreted as the sheaf of sections of germs of such functions.

BUT: it seems to me that this particular sheaf is also the sheaf of sections of a simpler bundle, namely the projection $X\times \mathbb R\to X$, because sections of this bundle are precisely continuous functions $X\to \mathbb R$.

Now, I know these two bundles aren't the same space, because, as Mac Lane and Moerdijk point out in Sheaves in Geomtry and Logic, the former (at least for $X=\mathbb R$) isn't Hausdorff, while the latter plainly is. Also, I seem to understand that such bundles of stalks are always étale, while the latter bundle is clearly not étale.

So, am I missing something, or do the sections of these two very different bundles produce the same sheaf?

Answer 1

These bundles have the same sections. But which is simpler depends on your point of view. $X\times \mathbb R\to X$ is simpler set theoretically but is `more complicated' topologically in the the sense that it is not a local homeomorphism. The sheaf, taken as a bundle, is a local homeomorphism.