91Ó°ÊÓ

\(X\) is a topological space and \(Y \subset X\) a subset with the subspace topology; prove that every closed subset of \(Y\) is of the form \(Y \cap V\) with \(V\) closed in \(X\).

Write down equations for a torus, a solid torus and a Möbius strip in terms of Cartesian coordinates \((x, y, z)\) or cylindrical polar coordinates \((r, \theta, z)\) for \(\mathbb{R}^{3}\). [Hint: you get a torus by rotating a circle about an axis outside it, and a Möbius strip by letting a diameter of the circle rotate simultaneously to get \(1,3,5, \ldots\) half-twists.]