A line through the origin is decided by any non-zero point on it, and two points decide the same line exactly when one is a multiple of the other. Taking the set of lines through the origin as a space is the projective construction, and it is a gluing: on the sphere, identify each point with the point opposite it. In one dimension the result is a circle again; in two it is a surface that does not fit inside ordinary space.

A compass needle without a marked tip points along a line rather than in a direction. The space of what such a needle can say is the projective line: every direction and the one opposite it are the same answer.

Each line through the origin meets the unit circle in two opposite points, and the projective line is what remains when each such pair is glued into one point. The picture is the instruction: identify $v$ with $-v$ on the sphere, in any dimension.

Real projective $n$-space, written $\mathbb{RP}^n$, is the set of lines through the origin in $\mathbb{R}^{n+1}$. Equivalently it is the sphere $$ with each point glued to the point opposite it, which is the description used here because a sphere is compact and the recognition theorem then applies.

• $\mathbb{RP}^1\cong S^1$: gluing opposite points of a circle gives a circle again, proved below by doubling the angle.
• $\mathbb{RP}^2$, the projective plane, is a compact connected surface that cannot be placed inside without crossing itself. The fact is stated here and belongs to a later course.

Describe the circle by an angle and double it: the point at angle t goes to the point at angle twice t. This map is continuous and onto, since doubling covers every angle as t runs half way round. Two points have the same image exactly when their angles differ by half a turn, which is exactly when they are opposite points of the circle. So the map glues exactly the antipodal pairs and nothing else. The circle is compact and Hausdorff, so the recognition theorem identifies the glued circle with the circle.

Double the angle of each point of the circle.

Doubling reaches every angle, and both coordinates are continuous.

Two angles double to the same angle exactly when they differ by half a turn.

The circle is closed and bounded in the plane, and a subspace of a metric space.

The recognition theorem identifies the glued circle with the circle.

Projective space is the space of lines through the origin, built as a sphere with antipodal points identified.

• Projective space is the space of lines through the origin.
• It is a sphere with each point glued to its opposite.
• $\mathbb{RP}^1$ is a circle, by doubling the angle.
• $\mathbb{RP}^2$ is a compact surface that does not fit inside $\mathbb{R}^3$.

Great work! You now know how to:

• describe projective space as lines through the origin;
• build it by gluing antipodal points of a sphere;
• prove that $\mathbb{RP}^1$ is a circle;
• name what is true of the projective plane and left unproved.