Metric on the antipodal sphere quotient ($d([u],[v])=\min (\Vert u-v\Vert ,\Vert u+v\Vert )$)

The displayed metric on antipodal classes of the unit sphere induces its quotient topology. Triangle inequality follows by choosing the signs that realize each minimum and applying the Euclidean triangle inequality. Distinct classes have disjoint saturated unions of small balls, proving the quotient is Hausdorff; compactness follows from the continuous quotient map.