$\Bbb S^3= \{(x,y,z,w)|x^2+y^2+z^2+w^2=1\}$ and $(x_1,y_1,z_1,w_1)\sim (x_2,y_2,z_2,w_2)$ iff $w_1>0, \ w_2>0$,
Consider $P: X \to X/{\sim~}$ (natural function). I know X is compact then $X/{\sim}$ is compact, and X is connected then $X/{\sim}$ is connected.
$\Bbb S^3$ is connected, but $\Bbb S^3/{\sim}$ is not connected.. i don't know what is wrong..(topology: subspace topology of usual top. on $\mathbb R^4$ and quotient top.)