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In an introductory topology class, one learns about covering maps $p : E \to B$. A natural first question to ask is if the composition of two covering maps is also a covering map, and the answer is no. The classical counterexample is the covering map involving a "countable series of hawaiian earrings" (see here), and in fact the only counterexample I've ever heard of.

However, to me it seems that the fact that the covering maps don't form a category should be a more substantial fact. I feel like there should be a better / more high-level explanation than "look at this crazy space!" So I suppose my questions are:

  • Is there some property of the counter-example in question that is key in the failing of the composability, and can this point us in the direction of other counter-examples?

  • Along the lines of the above, is there some kind of restriction we can make on the property of the spaces (or the maps) which does indeed make covering spaces into a category?

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