Math and stuff

Proposition Let $x_0, x_1 \in X$ be given. Then the relation of homotopy on paths from $x_0$ to $x_1$ is an equivalence relation. Solution Let $f$ be a path from $x_0$ to $x_1$. Consider $f_t(s) = f(s)$. $f_0 = f_1 = f$. For any $t \in [0, 1]$, $f_t(0) =...

Proposition Every non-constant polynomial with coefficients in $\mathbb{C}$ has a root in $\mathbb{C}$. Solution Let $p(z) = z^n + a_1z^{n - 1} + \cdots + a_n$ be given where $n \geq 0$. Suppose that $p(z)$ has no roots in $\mathbb{C}$. For each $r \geq 0$, define \[\begin{align*} f_r(s) = \frac{p(re^{2\pi...

Proposition A space $X$ is simply-connected iff there is a unique homotopy class of paths connecting any two points in $X$. Solution Suppose that $X$ is simply-connected. Let $x, y \in X$ be given. Since $X$ is path-connected, $x, y$ are connected by a path. Let $f, g$ be two...