Does there exist an uncountable set of subsets of the positive integers such that any two distinct subsets have finite intersection?
Solution
Answer: yes.
Consider first the set Q of rationals. For each positive real in the interval (0, 1), take a sequence of rationals converging to it. This gives an uncountable collection of subsets of Q. Clearly two collections have at most finitely many points in common. To get the corresponding subsets of the postive integers simply take a bijection between the positive integers and the rationals.
© John Scholes
jscholes@kalva.demon.co.uk
1 Jan 2001