Need urgent help with Advanced Maths assignment. Please see attachment for typed questions.

For each of the statements below determine whether it is true or false, providing reasons for your answer.

(a) For the real line R with the standard topology and its subset Q of all rational numbers we have Cl(Q) = R. [2 marks]

(b) For the real line R with the standard topology and its subset Q of all rational numbers we have ∂(Q) = R. [2 marks]

(c) Singleton sets always closed in Hausdorff spaces. [2 marks]

2. Let T = {(a,∞) : a ∈ [−∞,∞]}. (Note: when a = −∞ we have (a,∞) = R, while if a = ∞, then (a,∞) = ∅.)

(a) Show that T is a topology on R. [5 marks]

(b) Carefully explain whether T is Hausdorff or not. [3 marks]

3. Let X be a topological space and let K1,K2, . . . ,Km be compact subsets of X. Show that K = K1 ∪ K2 ∪ . . . ∪ Km is compact, too. [5 marks]