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We may use the representation of projective space via homogeneous coordinates to get a topological picture of these spaces.

- When we write
**R***P*^{1}=**R**^{2}- {(0, 0)}/~ with ~ as before, the equivalence classes are lines through the origin in**R**^{2}.

Each line meets the "upper semicircle" in a unique point except for the horizontal line which meets it twice.

Thus**R***P*^{1}is the space we get from the semi-circle by identifying its end points. This is a circle*S*^{1}.One can see the same thing by mapping the line to the circle by stereoscopic projection from the top point of the circle.

- The space
**R***P*^{2}is the set of lines through the origin in**R**^{3}-**0**. Each line meets the "upper hemisphere" in a unique point except for any horizontal line which meets it twice in opposite points on the equator. Thus can be made out of a hemisphere (topologically equivalent to a 2-dimensional disc) by identifying opposite points on the boundary.

This can be represented either by or by .Alternatively, each line in through the origin in

**R**^{3}-**0**meets the unit sphere*S*^{2}in a pair of antipodal points.

Thus**R***P*^{2}is the space we get from the sphere by identifying antipodal points.As a topological space,

**R***P*^{2}is*non-orientable*.

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