Math and stuff

Proposition Let $X$ and $Y$ be topological spaces. Show that if either $X$ or $Y$ is contractible, then every continuous map from $X$ to $Y$ is homotopic to a constant map. Solution Suppose $X$ is contractible. Let a continuous function $f: X \rightarrow Y$ be given. Then there exists a...

Proposition Suppose that $X$ is a topological space, and $g$ is any path in $X$ from $p$ to $q$. Let $\Phi_g: \pi_1(X, p) \rightarrow \pi_1(X, q)$ denote the group isomorphism induced by $g$. Show that if $h$ is another path in $X$ starting at $q$, then $\Phi_{g \cdot h} =...

Proposition Suppose $f, g: S^n \rightarrow S^n$ are continuous maps such that $f(x) \ne -g(x)$ for any $x \in S^n$. Prove that $f$ and $g$ are homotopic. Solution $S^n = \{ x \in \mathbb{R}^{n + 1} \mid \abs{x} = 1 \}$. Let continuous functions $f, g: S^n \rightarrow S^n$ be...

Proposition Show that the following are equivalent: $X$ is contractible. $X$ is homotopy equivalent to a one-point space. Each point of $X$ is a deformation retract of $X$. Solution $1 \rightarrow 2$ Let $X$ be a contractible space. Then there exists a point $p \in X$ and a continuous map...

Proposition Prove that the circle is not a retract of the closed disk. Solution Let $\overline{D} = \{ z \in \mathbb{C} \mid \abs{z} \leq 1 \}$ and $S^1 = \{ z \in \mathbb{C} \mid \abs{z} = 1 \}$. We will first show that $\overline{D}$ is simply connected. $\overline{D}$ is path-connected....