open set

In mathematics, an open set is a fundamental concept in topology, which is a generalization of Euclidean spaces. Instead of relying on distance, topology describes closeness and limits through relationships between sets. An open set is a collection of points within a larger set, defined by specific rules. These rules state that the intersection of any finite number of open sets must also be open, and the union of any number of open sets (finite or infinite) must also be open.1

The concept of open sets is crucial for defining continuity of functions. A function is considered continuous if, for any open set in the output space, the corresponding set of input values is also open. This abstract definition allows mathematicians to study properties of spaces and functions in a very general way, extending beyond the familiar geometry of Euclidean spaces.213