I know that $S^2$ is simply connected since by Sard's theorem any curve in $S^2$ ($1$ dimensional manifold) is a measure zero set hence it should be contained in $S^2-\{p\} $ for some $p\in S^2$ and $S^2-\{p\}\cong \Bbb{R}^2$ is simply connected.
But from this I cannot prove any two curves in $S^2$ has intersection $0$ mod $2$. Can anyone help me to complete the proof?