This is coming from my introduction to topology class.

"Let (Y, d') be a subspace of the metric space (X, d). Prove that a subset $\displaystyle O' \subset Y$ is an open subset of (Y, d') iff there is an open subset O of (X, d) such that $\displaystyle O'=Y \cap O$. Prove that a subset $\displaystyle F'=Y \subset F$. For a point $\displaystyle a \in Y$, prove that a subset $\displaystyle N' \subset Y$ is a neighbohood of a iff there is a neighborhood N of a in (X, d) such that $\displaystyle N' = Y \cap N.$"

How and where do I start with this? The definition of a open subset?

A subset O of a metric space is said to be open if O is a neighborhood of each of its points.