Borsuk-Ulam Theorem (idea) by terse

This would be true if indeed the Earth is a mathematically perfect sphere.

But it's not. The Earth is slightly squashed like a flattened tennis ball, bulging outwards at the equator due to the spin of magma. In addition, there are the small problems of geography, coastal dynamics, and ablative heat transform (I don't know what that means, but it sounds impressive).

The Earth's shape is also affected (minimally) by the position of the moon, and the other solar bodies.

So unless this is true for any vaguely sphere like surface, the theorem doesn't apply.

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