Prove that there can be at most countably many disjoint letter T's in the plane

Solution 1:

Given any collection of T's define a function that takes each T to three rational coordinates obtained from the superimposed ice cream cone. say (p,q,r) where p and q are on distinct sides of the cone and r is in the ice cream. If two disjoint T's mapped to the same first two coordinates (match in the cone) which can happen then the r components are distinct. To see this is the case note there is a restrictive angle less than $\pi$. Hence the function is injective and therefore the collection of T's is countable.

Solution 2:

Hint: if you have an uncountable collection, it contains a sequence with distinct centers in a compact subset and lengths of horizontal and vertical intervals bounded away from zero. Therefore....