Metric on the antipodal sphere quotient
= Metric on the antipodal sphere quotient
{title2=$d([u],[v])=\min(\|u-v\|,\|u+v\|)$}
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.