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In the book Differential Topology by Morris W. Hirsch, he claims:

Non-paracompact manifolds never appear naturally.

What are some example of non-paracompact manifolds (or spaces) that do appear naturally?

(The long line for example seems to only be used for counterexamples so it's irrelevant to this question)

Carla only proves trivial prop
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  • More generally, any nonmetrisable manifold must fail to be paracompact. – Tyrone Oct 19 '22 at 14:44
  • It is a bit vague to say that something does never appear naturally. However, have a look at https://math.stackexchange.com/q/98105 to understand the reasons for requiring paracompactness. – Paul Frost Oct 19 '22 at 16:44
  • @Tyrone But do those appear naturally? I'm looking at examples where this kind of space appears naturally. I am aware that appear naturally is a big vague that's why i added the tag soft-question. – Carla only proves trivial prop Oct 19 '22 at 18:03
  • What is the definition of "appearing naturally" ? – Ulli Oct 19 '22 at 18:44
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    @Mariano Suárez-Álvarez: So, why is the long line not appearing naturally? BTW, I researched in set-theoretic topology. We never talked about something appearing naturally. – Ulli Oct 20 '22 at 19:55
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    I recommend to read the excellent survey article "The Theory of Nonmetrizable Manifolds" by Peter Nyikos in the Handbook of Set-Theoretic Topology. In particular, section 3 contains a lot of examples, so everyone can decide by him-/herself whether one of these "appears naturally". – Ulli Oct 21 '22 at 10:11

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