I have a question about the definition of a homotopy between loops:
Let $\alpha$ and $\beta$ be loops with base point $x$ in a topological space $X$. A homotopy from $\alpha$ to $\beta$ is a continuous function $H$ from $[0,1]^2$ to $X$ such that $H(s,t)=f_s(t)$ where $f_s$ is a loop with base point $x$ and such that $f_0=\alpha$ and $f_1=\beta$.
What are the opens relative to $[0,1]^2$? If we're calling $H$ continuous, don't we need to know what topology on $[0,1]^2$ we're talking about? Is a certain topology implied?