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**).
> 
> 
>
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 $f(x)$.