is a neighbourhood of all points in The open sets then satisfy the axioms given below. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining to be a neighbourhood of if includes an open set such that Definition via open sets A topology on a set may be defined as a collection of subsets of , called open sets and satisfying the following axioms: The empty set and itself belong to Any arbitrary (finite or infinite) union of members of belongs to The intersection of any finite number of members of belongs to As this definition of a topology is the most commonly used, the set of the open sets is commonly called a topology on A subset is said to be in if its complement is an open set. Examples of topologies Given the trivial or topology on is the family consisting of only the two subsets of required by the axioms forms a topology of Given the family of six subsets of forms another topology of Given the discrete topology on is the power set of which is the family consisting of all possible subsets of In this case the topological space is called a discrete space. Given the set of integers, the family of all finite subsets of the integers plus itself is a topology, because (for example) the union of all finite sets not containing zero is not finite but is also not all of and so it cannot be in Definition via closed sets Using de Morgan's laws, the above axioms defining open sets become axioms defining closed sets: The empty set and are closed. The intersection of any collection of closed sets is also closed. The union of any finite number of closed sets is also closed. Using these axioms, another way to define a topological space is as a set together with a collection of closed subsets of Thus the sets in the topology are the closed sets, and their complements in are the open sets. Other definitions There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms. Another way to define a topological space is by using the Kuratowski closure axioms, which define