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

Answer: 
Let $F$ be a family of functions from a set $X$ to a space $Y$. $F$ might, for instance, be the set of all functions from $X$ to $Y$, or it might be the set of all continuous functions from $X$ to $Y$, if $X$ is a topological space. Each $f:X\to Y$ can be thought of as a point in the Cartesian product $Y^{|X|}$. To see this, for each $x\in X$ let $Y\_x$ be a copy of the space $Y$. Then a function $f:X\to Y$ corresponds to the point in $\prod\_{x\in X}Y\_x$ whose $x$-th coordinate is $f(x)$, and of course $\prod\_{x\in X}Y\_x$ is just the product of $|X|$ copies of $Y$, i.e., $Y^{|X|}$.

The product $Y^{|X|}$ is a topological space with the product topology; $F\subseteq Y^{|X|}$, so $F$ inherits a topology from the product topology on $Y^{|X|}$. This inherited topology is the *topology of pointwise convergence* on $F$.

It can easily be shown that a sequence $\langle f\_n:n\in\Bbb N\rangle$ in $F$ converges to some $f\in F$ in this topology if and only if for each $x\in X$, $\langle f\_n(x):n\in\Bbb N\rangle$ converges to $f(x)$ in $Y$. (More generally, a [net](http://en.wikipedia.org/wiki/Net_%28mathematics%29) $\langle f\_d:d\in D\rangle$ in $F$ converges to some $f\in F$ if and only if for each $x\in X$ the net $\langle f\_d(x):x\in D\rangle$ converges to $f(x)$ in $Y$.) This is the reason for the ***pointwise*** in the name.

Very often $Y$ is $\Bbb R^n$ or $\Bbb C^n$ for some $n$, and $X$ is some topological space. The topological structure of $X$ has no bearing on the topology of pointwise convergence, though it may help to determine the set $F$ of functions under consideration (e.g., the continuous ones).
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 $f(x)$.