noun Informally, a set such that the target point of a movement by a small amount in any direction from any point in the set is still in the set; exemplified by a full circle without its boundary.
noun A set which can be described as an (arbitrary) union of open balls. Equivalently, a set such that for every point in it, there is an open ball centered at that point, such that that open ball is contained by the set.
noun Most generally, a member of the topology of a given topological space.