General Topology/Definition, characterisations

Exercises

 * 1) Let $$X$$ be a set.
 * 2) Prove that $$\mathcal P(X)$$, the power set, is a topology on $$X$$ (it's called the discrete topology) and that when $$X$$ is equipped with this topology and $$f:X \to Y$$ is any function where $$Y$$ is a topological space, then $$f$$ is automatically continuous.
 * 3) Prove that $$\{\emptyset, X\}$$ is a topology on $$X$$ (called the trivial topology), and that when $$X$$ is equipped with this topology, $$Z$$ is any topological space and $$f: Z \to X$$ is any function, then $$f$$ is continuous.