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May 9, 2023 at 14:55 comment added Francesco Polizzi @BrianLins: well, when I wrote this example no precise definition was given. Anyway, I will leave the answer, which belongs to an enlarged version of the "other settings" category...
May 9, 2023 at 14:52 comment added Brian Lins Your manifold $M$ is not closed, but a real analytic variety (as defined belatedly above) would always be closed.
May 9, 2023 at 14:51 comment added Z. M Seemingly something easier is possible: $\arctan\colon\mathbb R\to(-1,1)$ seems to be a homeomorphism and real analytic?
May 9, 2023 at 13:25 history answered Francesco Polizzi CC BY-SA 4.0