## HW3 MA5311. Due February 15

Exercise. Let $X \subset \mathbf{R}^k$ be a subset homeomorphic to the closed ball $B_n \subset \mathbf{R}^n$.  Show that if $f: X \to X$ is continuous, then there exists $x \in X$ such that $f(x)=x$.

Exercise. Let $X$ be a one-dimensional compact manifold with boundary. Show that $X$ is diffeomorphic to a finite union of segments and circles (You may use the appendix in Milnor’s lecture notes).

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