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##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)##

As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in ##\mathbb R^2## subspace topology while the domain open interval is not, thus ##\beta## is not a smooth embedding.

Consider it from the point of view of "homeomorphism onto its image" definition, I was trying to find out an instance of image subset open in the subspace topology that actually is not open in the domain topology or the other way around.

Can you help me ? Thanks.