# What does it mean to say that a pair of points are antipodal in a topological sphere?

A pair of points are antipodal if they are diametrically opposite to each other. This definition makes perfect sense when one thinks of the unit 2-sphere centered at the origin and embedded in $R^3$; that is, the set of all points for which $x^2+y^2+z^2=1$.

However, what does it mean to say that a pair of points are antipodal in a topological sphere? If this question doesn't make sense, I fail to recognize when two points are antipodal when considering, say, an ellipsoid. For example, how does one make sense of the Borsuk–Ulam theorem for the ellipsoid?

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The meaning of "antipodal" depends on a choice of homeomorphism between an ellipsoid (or any other subset of $\mathbb{R}^n$ homeomorphic to a sphere) and the usual unit sphere. The Borsuk-Ulam theorem is true regardless of this choice.