Bornology

A bornological space is a type of mathematical structure, similar but not identical to a topological space. While a topological space deals with the notion of "closeness", a bornological space replaces this concept with "boundedness". Bornology, also known as bounded set theory, was developed to study the properties of "bounded" objects and operations, providing a framework to discuss what it means for a set to be "big" or "small".

Formally, a bornological space is a set $X$ along with a collection $B$ of subsets of $X$ (called the bornology or bounded sets) satisfying the following three properties:

1. Every subset of a bounded set is bounded: If $A \in B$ and $C \subseteq A$, then $C \in B$.

2. The union of a finite number of bounded sets is bounded: If $A_i \in B \forall i=1,2,...,n$, then $\bigcup A_i \in B$.

3. Every singleton set is bounded: ${x} \in B \forall x \in X$.

While bornological spaces share certain similarities with topological spaces, they should not be considered identical. The focus in a bornological space is the notion of boundedness, whereas in a topological space, the focus is on open sets and continuity. However, there are deep connections between the two. For instance, every metric space induces both a topology and a bornology. In these cases, concepts of continuity (a topological concept) and boundedness (a bornological concept) are closely intertwined.

You can think of a bornological space as a generalization of the concept of a metric space, allowing for discussions about the "size" of sets without a specific distance function. Bornological spaces find significant applications in the field of functional analysis, particularly in the study of spaces of functions and operators.