Problem 11. In this problem, you will prove a classification theore...

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Problem 11. In this problem, you will prove a classification theorem for 1-dimensional manifolds. In particular, you will prove that every closed 1-dimensional manifold (without boundary) is homeomorphic to the circle S¹. Let M be a closed (i.e., compact and connected) 1-dimensional manifold. (a) Show that M is can be covered by finitely many open sets U1 Un n each of which is homeomorphic to an open interval in R. (b) Prove that if hi : Ui (ai, bi) is a homeomorphism, then there is some Ei and some Ui different from Ui such that if dist( (x, {ai, bi}) < Ei, then hit (x) E Uj. (c) Pick a point p € U1. Create the path r in M which starts at p and "goes right" in (ai,bi). More specifically, (h1 or) (t) = h1 (p) +t. At some point r enters a different Ui. Show that can be extended to cover the entirety of the set Ui. (d) Iterate the above process by induction. (e) Prove that eventually the path returns to the point p, at which point we stop the path 7 (hint: there are only finitely many Ui). (f) Prove that every Ui is eventually traversed in its entirety by 7 (hint: connectedness). (g) Prove that the image of 7 is on one hand M, and on the other hand is homeomorphic to S¹ because it is a map from a closed interval which identifies only the end points.

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