Prove that the map $c:{\mathbb {T}}^{3}=S^{1}\times S^{1}\times S^{1}\setminus \Delta\longrightarrow \{\pm 1\}$ is continuous where $\Delta$ is the diagonal $\Delta=\{(g_i,g_j,g_k)\}$ for $i=j$ or $j=k$ or $i=k$ and $c$ has the following properties:
1)($\textit{Left-Invarianc}$): $c(ag_1,ag_2,ag_3)=c(g_1,g_2,g_3)$ for all $a,g_1,g_2,g_3\in S^1$ with $g_i\ne g_j$ for any $i$ and $j$
2)($\textit{Co-cycle condition}$): $c(g_1,g_2,g_3)-c(g_1,g_2,g_4)+c(g_1,g_3,g_4)-c(g_2,g_3,g_4)=0$ for all $g_1,g_2,g_3,g_4\in S^1$ with $g_i\ne g_j$ for any $i$ and $j$
These two conditions basically tells us that there is a certain order on circle which is left-invariant.
I want to prove or disprove the continuity of $c$ where $\{\pm 1\}$ is given discrete topology. We see that $\mathbb{T}^{3}=S^{1}\times S^{1}\times S^{1}$ is a three dimensional torus and not much of its topological facts are given in the wiki article. Please suggest where should I start.