Math and stuff

Proposition Let $K = \{ U_{\alpha} \}$ be an open cover of $I$. Then there exist $0 = s_0 < s_1 < \cdots < s_m = 1$ such that for each $i = 0, \cdots, m - 1$, there exists an $\alpha$ such that $[s_i, s_{i + 1}] \subset A_{\alpha}$....

Proposition Whenever $S^2$ is expressed as the union of three closed sets $A_1, A_2$, and $A_3$, then at least one of these sets must contain a pair of antipodal points $\{ x, -x \}$. Solution Since $S^2$ is nonempty, not all $A_1, A_2, A_3$ are empty. If only one of...

Proposition $\pi_1(X \times Y)$ is isomorphic to $\pi_1(X) \times \pi_1(Y)$ if $X$ and $Y$ are path-connected. Solution By Theorem 18.4 from Munkres, a function is continuous if and only if all of its coordinate functions are continuous. Let $(x_0, y_0) \in (X \times Y)$. Let $\phi: \pi_1(X \times Y, (x_0,...

Proposition Every continuous map $h: D^2 \rightarrow D^2$ has a fixed point, that is, a point $x \in D^2$ with $h(x) = x$. Solution Suppose that $h(x) \ne x$ for any $x \in D^2 = \{ x \in \mathbb{R}^2 \mid \abs{x} \leq 1 \}$. Define $r: D^2 \rightarrow S^1$ as...

Proposition Let $x_0, x_1 \in \mathbb{R}^n$ be given. Let $f_0, f_1$ be two paths from $x_0$ to $x_1$ in $\mathbb{R}^n$. Then $f_0$ and $f_1$ are homotopic. Solution Consider the family of functions $f_t(s) = (1 - t)f_0(s) + tf_1(s)$. We claim that it is a homotopy of $f_0$ and $f_1$....