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Let $X$, $Y$ be closed subsets of a topological space $A$ such that $A = X \cup Y$, and let $B$ also be a topological space. Show that if $f: A \to B$ is continuous when restricted to both $X$ and $Y$, then $f$ is continuous.
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Let $B$ be an open ball in $\mathbb{R^n}$. Show that any path $\alpha$ in $B$ is path-homotopic in $B$ to a straight line path $\beta$, i.e, with $\beta: s \mapsto x + s\cdot y$, for some points $x, y\in\mathbb{R}^n$.
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Let $U\subset \mathbb{R^n}$ be an open set and $\alpha:[0, 1]\to U$ be a path. Show that there is an integer $m>0$ and a collection of open balls $B_i$, $0\leq i<m$, with $B_i\subset U$, so that for $i=0, 1,\dots, m-1$, $\alpha([i/m, (i+1)/m])\subset B_i$.
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Let $\alpha:[0,1]\to X$ be a path in a topological space $X$ and let $m>0$ be an integer. For $0\leq i < m$, define a path $\alpha_i: [0, 1]\to X$ by Show that $\alpha$ is path-homotopic in $X$ to $\alpha_0* \alpha_1 * \dots * \alpha_{m-1}$.
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Let $U\subset \mathbb{R^n}$ be an open set and $\alpha:[0, 1]\to U$ be a path. Show that there are paths $\alpha_1$, $\alpha_2$, …, $\alpha_m$ so that
- $\alpha$ is path-homotopic in $U$ to $\alpha_1* \alpha_2 * \dots * \alpha_m$.
- each path $\alpha_i$ is path-homotopic in $U$ to a straight line path $\beta_i$.
Thus, $\alpha$ is path-homotopic to a polygonal path.