Maths assignment

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

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]

4. Let X be a topological space. Prove that Int(A ∩ B) = Int(A) ∩ Int(B) for all subsets A and B of X. [8 marks]

5. Let (X, T ) and (Y, S) be two topological spaces and f, g : X → Y be two continuous maps. Show that, if (Y, S) is Hausdorff, the set Υ = {x ∈ X : f(x) ̸= g(x)} is open. [5 marks]

6. Let (X, T ) be a Hausdorff topological space and let K1 ⊇ K2 ⊇ . . . ⊇ Kn ⊇ . . .be an infinite sequence of non-empty compact subsets of X. Show that Kn ̸= ∅. i.e. there exists a point x ∈ X such that x ∈ Kn for all n ≥ 1. [8 marks]