Complex Analysis/The compact-open topology

${\mathfrak {S}}$-convergence and definition

What I will write on ${\mathfrak {S}}$-convergence will require knowledge of uniform structures as taught by Bourbaki's general topology book. It will not be necessary to understand the concept in order to understand anything else in the book.

Let $S$ be any set and $X$ a uniform space. Let $A\subset S$ be a subset of $S$. We consider the set of functions from $S$ to $X$; we may denote it by $X^{S}$. Assume we are given an entourage $V$ of $X$. We may then define the set of all pairs functions $(f,g)$ contained in the set $X^{S}\times X^{S}$ which have the property that for all $x\in A$, we have $(f(x),g(x))\in V$; this set will be denoted by

$W(A,V)$.

In fact, as $V$ ranges over a fundamental system of entourages of $X$, the sets $W(A,V)$ form a fundamental system of entourages on $X^{S}$, and the topology induced by the corresponding uniform structure is called the topology of uniform convergence on $A$.

Now suppose that we have a family of subsets $(A_{i})_{i\in I}$ of $X$; we shall call it, as it is customary, ${\mathfrak {S}}:=\{A_{i}|i\in I\}$. For each $i$, we may form the topology of uniform convergence of $A_{i}$ as above; for each $i$ will result a topology on $X^{S}$. Then we may form the least upper bound topology of these topologies; this is what's called the topology of ${\mathfrak {S}}$-convergence.

The compact-open topology is a special case of this construction; let $S$ be a topological space, and take ${\mathfrak {S}}$ to be the set of all compact subsets of $S$. The topology of ${\mathfrak {S}}$-convergence for this situation is called the compact-open topology. We now write this in a fashion so that everyone, even those who are not familiar with uniform spaces, will be able to understand the definition:

Definition 7.1:

Normal families

Definition 7.2:

Let ${\mathcal {F}}$ be a family of functions whose domain of definition is a metric space $(M,d)$

A general Arzelà–Ascoli theorem

The classical Arzelà–Ascoli theorem is a well-known theorem in analysis. It states that whenever we have a bounded, equicontinuous family of functions defined on a compact set, this family will constitute a relatively compact set