A topological space (or simply a topology) is an pair ), where is a set and is a subset of the power set of satisfying the following relations:
- Both and the empty set are elements of ;
- The union of the elements of any subset of is an element of ;
- The intersection of finitely many elements of is an element of .
By abuse of language, the topology is called simply "the topology " or "the topology ", depending on context.
The elements of are usually called open sets. The subsets of whose complements are elements of are then called closed sets.
It follows from the principles of set theory that the intersection of any family of closed sets is closed, and that the intersection of any family of open sets is open.
When checking whether a set is a topology, by induction it suffices to show that the intersection of any two elements of is also an element of .
Trivially, the sets , and (the power set of ) are both topologies. They are the least and greatest topologies on , respectively.
If is a family of topologies on , then so is their intersection, .
Let be a collection of subsets of . Since is a topology on of which is a subset, the set of topologies on that contain is not empty. It then follows from the previous paragraph that there exists a least topology on containing . This is the topology generated by .
Theorem. Let be a set with the property: for any in and any , there exists an element of such that Then the topology generated by is the set containing , and the subsets of with the following property: for every element of , there exists an element of such that and .
Proof. Call the set described in the theorem . From the properties of topologies it is evident that every topology containing must also contain . Hence it suffices to show that is in fact a topology.
By construction, is an element of ; since has no elements, it vacuously satisfies the theorem's condition and hence is also an element of .
Let be a family of elements of . If one of them is , then their union is , which is an element of . Otherwise, we note that every element of is an element of ; hence there exists some for which
Finally, suppose and are two elements of . If one, say , is equal to , then , and we are done. Otherwise, for every element of , there exist sets and for which . By hypothesis, there exists a set in such that This proves that . Hence is a topology, as desired.
In a metric space , we can define an open set to be a set such that for every element there exists an open ball centered at that is contained in . This is the topology generated by the neighborhoods of , and it is called the metric topology of . The closed sets of are then those sets such that every limit point of is an element of .
In , the topology generated by the infinite arithmetic sequences yields an interesting proof of the infinitude of primes. Specifically, consider the topology generated by the residue classes modulo , as ranges through all nonnegative integers. Then for every integer , the set is closed and open. If there are finitely many primes , it then follows that the union of all sets is closed. Since every integer except is divisible by some prime, it follows that the set is open, a contradiction.
Let be a set with a topology , and let be a subset of . The topology induces a topology on , namely the set of sets of the form , for . This topology is called the topological subspace .
Note that open and closed sets in the subspace are not necessarily open or closed in the space .