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> [!proposition] Proposition. ([[degree is sum of local degrees]])
> Suppose $f:\mathbb{S}^{d} \to \mathbb{S}^{d}$ such that $f^{-1}(\{ y \})=\{ x_{1},\dots,x_{k} \}$ is finite. Pick disjoint [[neighborhood|neighborhoods]] $U_{i}\ni x_{i}$, $U_{i} \subset \mathbb{S}^{d}$ open, such that each $f(U_{i}) \subset V$ for some open neighborhood $V \ni y$. Then the [[degree of a continuous map on the sphere|degree]] of $f$ equals the sum of [[local degree|local degrees]]: $\text{deg }f = \sum_{i=1}^{n}\text{deg}(f) |_{x_{i}}.$
^proposition
> [!basicexample]
The most intuitive example is if $f:\mathbb{S}^{n} \to \mathbb{S}^{n}$ is [[covering space|covering map]]. E.g. if $n=1$, the proposition says that 'to compute the [[degree of a continuous map on the sphere|winding number]] of $f$, count how many times (with sign) $f$ winds around $\mathbb{S}^{1}