91Ó°ÊÓ

(h) Suppose p: \(\tilde{\mathrm{X}} \rightarrow \mathrm{X}\) is a covering map with \(\mathrm{X}\) path connected. Prove that the cardinal number of \(p^{-1}(x)\) is independent of \(x \in X\). If this number is \(n\) then we say that \(p: \widetilde{X} \rightarrow X\) is an \(n\)-fold covering.

(i) Find a two-fold covering \(p: \mathrm{S}^{1} \times \mathbf{S}^{1} \rightarrow \mathrm{K}\) where \(\mathrm{K}\) is the Klein bottle.

(j) A subset \(\Sigma\) of a space is a simple closed curve if it is homeomorphic to \(\mathrm{S}^{1}\). Let \(\mathrm{p}: \mathrm{S}^{2} \rightarrow \mathrm{R} \mathrm{P}^{2}\) be the canonical projection of the sphere onto the projective plane. Prove that if \(\Sigma\) is a simple closed curve in \(\mathbf{R} \mathrm{P}^{2}\) then \(\mathrm{p}^{-1}(\Sigma)\) is either a simple closed curve in \(\mathrm{S}^{2}\) or is a union of two disjoint simple closed curves. (Ilint: Consider \(\Sigma\) as the image of a closed path in \(R P^{2}\).

(m) Does there exist a topological space \(Y\) such that \(S^{1} \times Y\) is homeomorphic to \(\mathrm{R} \mathrm{P}^{2}\) or to \(\mathrm{S}^{2}\) ?

(n) Suppose that \(\mathrm{p}: \mathrm{X} \rightarrow \mathrm{Y}\) is a covering map and that \(\mathrm{X}, \mathrm{Y}\) are both Hausdorff spaces. Prove that \(\mathrm{X}\) is an \(\mathrm{n}\)-manifold if and only if \(\mathrm{Y}\) is an n-manifold.

(n) Suppose that \(\mathrm{p}: \mathrm{X} \rightarrow \mathrm{Y}\) is a covering map and that \(\mathrm{X}, \mathrm{Y}\) are both Hausdorff spaces. Prove that \(\mathrm{X}\) is an \(\mathrm{n}\)-manifold if and only if \(\mathrm{Y}\) is an n-manifold.

(p) Let \(\mathrm{p}: \tilde{\mathrm{X}} \rightarrow \mathrm{X}\) be a covering and let \(\mathrm{f}, \mathrm{g}: \mathrm{Y} \rightarrow \widetilde{\mathrm{X}}\) be two continuous maps with \(\mathrm{pf}=\mathrm{pg}\). Prove that the set of points in \(\mathrm{Y}\) for which \(\mathrm{f}\) and \(\mathrm{g}\) agree is an open and closeá subset of \(\mathrm{Y}\).