Some Algebra of Dimensions

Euclidean Intersections

Suppose Q and R are lines in the plane.

Typically, Q and R do intersect in a point.

How do their codimensions add?

(2 - dim(Q)) + (2 - dim(R)) = (2 - 1) + (2 - 1) = 2

= 2 - 0 = 2 - dim(Q ∩ R)

That dim(Q ∩ R) = 0 reinforces the observation that Q and R intersect in a point, which has dimension 0.

Suppose P is a point in the plane and Q is a line in the plane.

Typically, P and Q do not intersect.

How do their codimensions add?

(2 - dim(P)) + (2 - dim(Q)) = (2 - 0) + (2 - 1) = 3

= 2 - (-1) = 2 - dim(P ∩ Q)

That dim(P ∩ Q) = -1 expresses the condition that typically P and Q do not intersect.

From these examples we hypothesize that if A and B lie in n-dimensional space,

n - dim(A ∩ B) = (n - dim(A)) + (n - dim(B))

That is,

codim(A ∩ B) = codim(A) + codim(B)

Exercise: verify this rule for the intersections of lines, planes, and points in 3-dimensional space.

Return to the intersection of sets.