An open set is a set in which all the points in the set are "interior" points, meaning that they are not on the boundary of the set. A closed set, on the other hand, contains all its boundary points.
A subset of R is an open set if every point in the set has a neighborhood (a set containing all points within a certain distance from the original point) that is also contained in the set.
A function is continuous if and only if the inverse image of an open set is also open. In other words, the preimage of an open set is an open set.
To prove that a subset of R is not an open set, you can find a point in the set that does not have a neighborhood contained in the set. This would violate the definition of an open set.
Open sets are a fundamental concept in topology, as they help define the topology of a space. The collection of all open sets in a space is called the topology of that space. Open sets also allow for the definition of other important concepts in topology, such as continuity, convergence, and compactness.