(a) Let X be a set and \(\mathcal{T}\) a collection of subsets of X. State three properties that \(\mathcal{T}\) must satisfy for the pair \((X, \mathcal{T})\) to be considered a topological space?
[10 Marks]
(b) Consider the set \([-1, 1]\) and a collection, \(\mathcal{T}\), of subsets of \([-1, 1]\), where \(U \in \mathcal{T}\) if and only if \(0 \notin U\) or \((-1, 1) \subseteq U\). Prove that \(\mathcal{T}\) is a topology on \([-1, 1]\). This topology is known as the either-or topology. [10 Marks]
(c) In the space \(([-1, 1], \mathcal{T})\), calculate
(i) \((0, 1/2)\),
(ii) \((-1/2, 1/2]°) \),
(iii) \(\partial(-1, 0)\).
[10 Marks]
