SECTION 4.1 Basics of Set Theory 205
- We saw on page 141 that the complete graphK 5 cannot be planar,
i.e., cannot be drawn in the plane. Let’s see if we can draw it
elsewhere. Start by letting
C = {(x,y)∈R^2 |x^2 +y^2 ≤ 1 };therefore, C is just the “disk” in the plane with radius 1. We
define an equivalence relationRon C by specifying that a point
(x,y) on the boundary ofC (so x^2 +y^2 = 1) is equivalent with
its “antipodal” point (−x,−y). Points on the interior of C are
equivalent only with themselves. We call the quotient setC/Rthe
real projective plane, often writtenRP^2. (Recall thatC/Ris
just the set of equivalence classes.)Explain how the drawing to the
right can be interpreted as a draw-
ing of K 5 in the real projective
plane. Also, compute the Euler
characteristicv−e+ffor this draw-
ing.- Here’s another construction ofRP^2 , the real projective plane; see
Exercise 10, above. Namely, take the unit sphere S^2 ⊆ R^3 , de-
fined by S^2 = {(x,y,z) ∈ {R^3 |x^2 +y^2 +z^2 = 1}. We define a
“geometry” on S^2 by defining points to be the usual points on
the sphere and defininglinesto be “great circles,” i.e., circles on
the sphere which form the shortest path between any two distinct
points on such a circle. Notice, therefore, that the equator on the
earth (approximately a sphere) is a great circle; so are the latitude
lines. With this definition of lines and points we see that Euclid’s
parallel postulate^6 is violated as distinct parallel lines simply don’t
exist: any pair of distinct lines must meet in exactly two points.
(^6) viz., that through any line1 and any pointPnot on 1 there is a unique line2 through the pointPand not intersecting 1.