[nyrikki]

I should add that “connectedness” is a local property that has to be everywhere.

The antipode pairs don’t have unique identities they are identified together, but IIRC they are never close enough to break that local condition of connectivity.

You could use the unit circle and a line at x=2, connecting points of that line through the origin and see how that identification of two points on the unit circle happens with that one line.

That may help but it wouldn’t explain the issues with so(3) that results in a 720 deg rotation being equivalent to a 0 deg rotation.

It would explain the point at infinity concepts as the poles are indescribable with a line from X=2 through the origin.