Brain Teaser: Connecting Points in the Depleted Euclidean Plane
© 1999 Edward G. Rozycki, Ed.D.

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Partition the real plane into three subsets:

R, rational points - the set of all points, (Xr,Yr), both coordinates rational;

I, irrational points - the set of all points, (Xi,Yi), both coordinates irrational;

Q, quasi-rational points - the set of all points, (Xq,Yq), one and only one coordinate rational.

Problem:
a. show that in the depleted plane consisting of (I union Q), i.e. the plane with the rational lattice removed, any two points can be connected by a continuous path. (Indeed, the path may consist of only two connected straight line segments.)

b. Consequently, show also that any two points in the complete plane, (I unionQ unionR), may be connected by a path crossing no other rational points.

Hint: Do not argue that between any two points on a straight line there must be a rational point. This is easily demonstrated to be false. The proof follows readily from this last consideration.