Show that if $X$ is path-connected, then every path $\alpha :I \to X$ are homotopic with each other (neglecting the condition rel $\{0,1\})$.
Since $X$ is path-connected we have that for every $a,b \in X$ there exists $\beta :I \to X$ such that $\beta(0)=a$ and $\beta(1)=b$.
Now for paths $\alpha, \gamma : I \to X$ what I need to find is a continuous map $h:I^2 \to X$ such that $$h(x,0)=\alpha(x) \text{ and } h(x,1)=\gamma(x).$$
How is the path-connectedness condition helpful here? I don't see how I can use it. I suppose I somehow need to take into account the fact that I can join every two points with paths when considering the homotopy, but I cannot figure out how.
Best Answer
To solve this problem you want to take advantage of a very important fact about homotopy, namely it is an equivalence relation.
Knowing that, suppose you can also prove the following two things:
So, let $f, g : [0,1] \to X$ be any two paths in a path connected space $X$.
You know from item 1 that there exists a constant path $P : [0,1] \to X$ such that
You also know from item 1 that $g$ is homotopic to some constant path that I'll denote $Q : [0,1] \to X$. Applying the symmetry law, it follows that
You also know from item 2 that
Since $f$ is homotopic to $P$, and $P$ is homotopic to $Q$, and $Q$ is homotopic to $g$, you may conclude from the transitive law that $f$ is homotopic to $g$.