General Topology/Definition, characterisations

Exercises

1. Let $X$ be a set.
   1. Prove that $𝒫(X)$, 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.
   2. Prove that $\{\mathrm{\varnothing },X\}$ 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.