You mention that covering spaces are étalé spaces but that not every étalé space is a covering space. A nice (and important) exercise is to identify which sheaves correspond to covering spaces. Spoiler: locally constant
You can turn a bundle into a presheaf, but you get a very nice kind of presheaf called a sheaf. You can turn a presheaf into a bundle, but you get a very nice kind of bundle called an etale space. Sheaves and etale spaces are secretly the same!https://johncarlosbaez.wordpress.com/2020/01/07/topos-theory-part-2/ …
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Nice! And yes, important! I drew a lot of bad pictures in an attempt to help students visualize etale spaces that aren't covering spaces.
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This sounds like cooking... now i am hungry
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