Let X={ a,b,c}.

(I) Find all topological spaces on X . Try to use symmetry in your argument,instead of just exhausting all cases. Try to find some general rules so you may generalize them .

(ii ) Provide the Hasse diagram for those topological spaces under the set containment relation .

( iii) Find the equivalent homeomorphic classes of all topological spaces on X and the corresponding Hasse diagram .

(iv) Tell some of the findings, the more the better. ( extra credit)

SOLVE THE ABOVE PROBLEM WITH ALL NECESSARY STEPS IN DETAIL.