Question: What is the Topology of point-wise convergence? It has been stated in lectures but I am unfamiliar with it.

Answer: 
The topology of point-wise convergence is a topology on a function space. Let $Z=Y^X$ be the space of functions from $X$ to $Y$. Then a sequence $(f\_i)\_{i\in \mathbb N}$ converges to a function $f$ if for all $x\in X$, the sequence $(f\_i(x))$ converges to $f(x)$.
It's a topology on the functions of a space such that a sequence of functions $\phi\_n$ converges to $\phi$ if and only if for all $x$ in the space the sequence of points $\phi\_n(x)$ converges to $\phi(x)$.