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The Hairy Ball Theorem states that every vector field on $S^2$ has a zero, unlike the circle $S^1$ which has a nowhere-zero vector field, usually written $\frac{\partial}{\partial\theta}$. So a natural question to ask is which spheres $S^n$ have a nowhere-zero vector field? Given the observations so far, one might guess that it depends on the parity of $n$ (...
You need some more assumptions in order to get ahead. For example, if $X$ is compact, then $C_0(X)=C(X)$. Indeed, $$C_0(X)\equiv \{f\in C(X)\,|\,\forall\varepsilon\in\mathbb{R}_{>0},\,f^{-1}\left(\mathbb{C}-B_{\varepsilon}\left(0_\mathbb{C}\right)\right)\in Compact\left(X\right)\}$$ However, if $f$ is continuous, since \$\mathbb{C}-B_{\varepsilon}\left(...