Topology of point wise convergence

Question

I encountered this topology in the exercise below. I was wondering if there is any connection to algebraic geometry? This looks like algebraic geometry but instead of points we are dealing with functions.

Answer 1

No, this has nothing to do with Algebraic Geometry. It is a way of defining a topology in the space of all functions from a metric space $X$ into a metric space $Y$ such that a sequence $(f_n)_{n\in\mathbb N}$ of functions from $X$ into $Y$ converges to a function $f\colon X\longrightarrow Y$ if and only if$$(\forall x\in X):\lim_{n\to\infty}f_n(x)=f(x).$$