I was given the following definition for "simple connectedness":
"Let $X$ be an arcwise connected, locally arcwise connected space. Then, X is simply connected if its fundamental group is trivial, or equivalently, if every closed path in $X$ is homotopic to a constant."
I seek to prove this "equivalence" claim.
Let's assume that every closed path in $X$ is homotopic to a constant, $e_x$. (Recall that $e_x(t) = x$, where $x \in X$ and $t$ is on some interval). Let $\alpha$ and $\beta$ be homotopic to $e_x$. That is, $\alpha \simeq e_x$ and $\beta \simeq e_x$. Since homotopy is an equivalence relation, it follows that $\alpha \simeq \beta$. Therefore, $\alpha \in <\beta>$, where $<\beta>$ is the homotopy equivalance class of $\beta$. Hence, $<\beta>$ is nonempty. Now, the fundamental group of $X$ is the group of homotopy equivalence classes of loops in $X$. My question is this: How can the fundamental group be trivial, since it contains $<\beta>$?
Thanks in advance.