Fourth Homework Set Answers

7. For A and B in 2-dimensional space, the codimension formula

codim(A ∩ B) = codim(A) + codim(B)

becomes

dim(A ∩ B) = dim(A) + dim(B) - 2

With dim(A) = 1.585 this becomes dim(A ∩ B) = -0.415 + dim(B).

If dim(B) < 0.415, then dim(A ∩ B) < 0 and so typically A and B do not intersect.

If dim(B) = 0.415, then dim(A ∩ B) = 0 and so typically A and B intersect in isolated points.

So 0.415 is the smallest value of dim(B) for which we expect A and B to have a non-empty intersection.

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