Math and stuff

Proposition Prove the following facts about retracts: A retract of a connected space is connected. A retract of a compact space is compact. A retract of a retract is a retract; that is, if $A \subset B \subset X$, $A$ is a retract of $B$, and $B$ is a retract...

Proposition The path homotopy relation is preserved by composition with continuous maps. That is, if $f_0, f_1: I \rightarrow X$ are path-homotopic and $\phi:X \rightarrow Y$ is continuous, then $\phi \circ f_0$ and $\phi \circ f_1$ are path-homotopic. Solution Suppose $f_0$ and $f_1$ are paths from $x_0$ to $x_1$, and...

Proposition Let $X$ be a path-connected topological space. Let $f, g: I \rightarrow X$ be two paths from $p$ to $q$. Show that $f \sim g$ if and only if $f \circ \overline{g} \sim c_p$. Show that $X$ is simply connected if and only if any two paths in $X$...

Proposition Let $X$ be a topological space. For any points $p, q \in X$, path homotopy is an equivalence relation on the set of all paths in $X$ from $p$ to $q$. Solution Let $f, g, h$ be paths in $X$ from $p$ to $q$. Reflexive property $F(s, t) =...

Proposition Let $B \subset \mathbb{R}^n$ be any convex set, $X$ be any topological space, and $A$ be any subset of $X$. Show that any two continuous maps $f, g: X \rightarrow B$ that agree on $A$ are homotopic relative to $A$. Solution We claim that the straight-line homotopy $F(x, t)...