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.
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The real projective plane, often denoted as \(\mathbb{RP}^2\), is a two-dimensional manifold that captures the idea of lines through the origin in three-dimensional space. Here are some key concepts to understand what the real projective plane is: 1. **Definition**: The real projective plane can be defined as the set of all lines through the origin in \(\mathbb{R}^3\).
For some reason, Ciro Santilli is mildly obsessed with understanding and visualizing the real projective plane.
To see why this is called a plane, move he center of the sphere to , and project each line passing on the center of the sphere on the x-y plane. This works for all points of the sphere, except those at the equator . Those are the points at infinity. Note that there is one such point at infinity for each direction in the x-y plane.