This exercise is from a "challenge" list that my analysis teacher gave us to do:
Let $X$ be a set. There are two trivial topologies:
Indiscrete (not sure if this is the actual name, as I am translating from another language) Topology: the open sets are exactly $X$ and the empty set.
Discrete Topology: every subset of $X$ is open.
Deduce that the indiscrete topology is not metrizable and the discrete topology is metrizable.
My problem is that this is a "first/second" course in real analysis and no one in the class has seen topology and metric spaces beyond the basics, so I pretty much have no idea of how to do this. Suggestions on how to prove this (proofs itself are not needed) will be very much appreciated.
P.S.: I've looked myself at some topology books and I think I'm not supposed to use anything like Hausdorff spaces and stuff like that (not even sure if that is the actual way to go).