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?


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.


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