Which of the following subsets is NOT a connected subset of real numbers (ℜ)?

Find the collection of subsets which is NOT a basis for any topology on X = {a, b, c}.

Let X = {a, b, c} and τ1, τ2, τ3, and τ4 be collections of subsets of X. Which of the following is NOT a topology on X?

Let X = ℝ2 be the Euclidean metric space, B1 be the collection of all open balls in ℝ2, and B2 be the collection of all open rectangular regions in ℝ2. If τ1 and τ2 are the topologies generated by B1 and B2 respectively, then which of the following is true?