What is the Topology of point-wise convergence? What is the Topology of point-wise convergence? It has been stated in lectures but I am unfamiliar with it.
 A: 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 $\langle f_d:d\in D\rangle$ 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).
A: 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 $f(x)$.
A: 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 $\phi(x)$.
A: 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).

