Concept guide
Topological space: fully verifying a chain of open sets on a three-point set, and recognising non-topologies
Let X={a,b,c} with a, b, c distinct. (1) Prove that τ₁={∅,{a},{a,b},X} is a topology on X, covering all 16 subfamilies (including the empty one); (2) prove that τ₂={∅,{a},{b},X} is not a topology, and say which checks it does pass; (3) compare τ₁ with the indiscrete and discrete topologies; (4) on ℕ, prove that τ_fin={finite subsets of ℕ}∪{ℕ} is closed under pairwise unions and finite intersections but is not a topology.