Definition: An open set is a set of points in a topological space that can be "separated" from its complement by a finite number of open sets. This means that, for any point in the open set, there exists an open set that contains that point and is completely contained within the original open set.

Properties: Open sets have several important properties that distinguish them from closed sets (sets that are complements of open sets). For example, open sets are often defined to be "dense" in a space, meaning that any point in the space can be approximated arbitrarily closely by a point in the open set. Open sets are also "unbounded" in the sense that they do not have a fixed boundary or edge.

Examples: Some common examples of open sets in topological spaces include the set of all real numbers, the set of all points in a plane that are not on a given line, and the set of all points in a 3-dimensional space that are not on a given plane.

Importance: Open sets are an important concept in topology because they provide a way to describe the local structure of a space. They are used to define important topological notions such as continuity and connectedness, and they are often used in the definition of other important sets such as neighborhoods and relatively open sets.