glad i have enough time this evening to glance my old differential geometry notes, consulting on basic topics and here's my self-made homework you can consult any textbook on differential geometry
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to have an open ball instead of a closed one so that all points it contains would be that of a manifold? open interval never really touching the boundary
https://en.wikipedia.org/wiki/Unit_sphere#Unit_spheres_and_balls_in_Euclidean_space
