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

Answer: 
From Munkres' *Topology, Second Edition*, $\S$46:

> 
> ***Definition.***   Given a point $x$ of the set $X$ and an open set $U$ of the space $Y$, let
> $$S(x,U)=\{f\mid f\in Y^X\hbox{ and }\ f(x)\in U\}.$$
> The sets $S(x,U)$ are a subbasis for *a* topology on $Y^X$, which is called the **topology of pointwise convergence** (or the **point-open topology**).
> 
> 
>
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 $\phi(x)$.