Question

For each of the following sets X and collections T of open subsets decide whether the pair X, T satisfies the axioms of a topological space. If it does, determine the connected components of X. If it is not a topological space then exhibit one axiom that fails.

(a) X = {1, 2, 3, 4} and T = {∅, {1}, {1, 2}, {2, 3}, {1, 2, 3}, {1, 2, 3, 4}}.

(b) X = {1, 2, 3, 4} and T = {∅, {1}, {2}, {3}, {4}, {1, 2, 3}, {1, 2, 3, 4}}.

(c) X = R and a subset U ⊂ R is open if and only if U is infinite or U = ∅.

(d) X = R and a subset U ⊂ R is open if and only if R \ U is countable. (A set is countable if, and only if, it is finite or in bijection with Z.)

Answer 1