Consider the two dimensional sphere $S^2$. It is obviously a two dimensional topological manifold without boundary.

Can one say that $S^2$ is a $3$-dimensional manifold $M$ with boundary such that $\text{int}(M)=\emptyset$ and $\partial M=S^2$. Does this description have sense?

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A 3-manifold with boundary is one in which every point has a neighborhood homeomorphic to either R^3 or a half space in R^3. Since this isn't true for the points on the sphere, it doesn't qualify.

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