Definition P4.1. A topology on a set is a collection of subsets of having the following properties:
and are in .
The union of the elements of any subcollection of is in .
The intersection of the elements of any finite subcollection of is in .
(A topology together with its set is written .)
Note. A set for which a topology has been specified is called a topological space.
Let and . Then we say is an open set.
On a topological space, finite intersections of open sets are open.
Example P4.2. Let be any three-element set . The following are topologies on :
And so on. Once the list is complete, you can get all topologies on by permuting ,, and .
The discrete topology is , the collection of all subsets of ; the trivial (or indiscrete) topology on is .
Definition P4.3 (The finite complement topology). Let be a set and let be the collection of all subsets such that either is finite or is all of .
Both and are in , since is finite and is all of . If is an indexed family of nonempty elements of , to show that is in we compute
which is finite because each set is finite. If are nonempty elements of , to show that we compute
the latter set being finite since it is a finite union of finite sets.
Comparing topologies
Definition P4.4. Suppose that and are two topologies on a given set . If , we say that is finer than . If properly contains , we say that is strictly finer than . We also say that is coarser than , or strictly coarser, in these two respective situations. We say is comparable with if either or .
Another term that is used, though less vivid, is larger and smaller, respectively. Rarely, the terms “weak” and “strong” topologies are invoked for the same purpose.
Bases
Definition P4.5. If is a set, a basis for a topology on is a collection of subsets of (called basis elements) such that:
For all , there is such that .
If belongs to the intersection of two basis elements and , then there is a containing such that .
Let us restate the basis in words of our own: ”A basis is a collection of sets in such that for all , there exists a basis element such that ; furthermore, for any with as basis elements, there is another basis element such that and .”
It naturally follows, then, that the topology generated by the basis is defined as follows: a subset is said to be open in (i.e. ) if for all there is such that and . This is to say that we can generate any element in the topology only using basis elements.
Of note: the collection of all unions of basis elements is exactly the topology on the set.
Math Major:
Open intervals in open sets in open sets in a topological space.
Open interval:.
For any point in an open interval, we can find another interval, completely contained in the original interval, centered at .
Proposition P4.6 (Claim). For all , there is such that .
Proof. Given , let . Take some ; we want to show . Observe that and , so . Notice , so
∎
Definition P4.7. A set is open if for all there is such that .
Example P4.8. The following sets are open in :
(a union of open sets, so open)
Show solution
: given , note that for all ,.
: vacuously true.
: take ; for all such ,.
: same.
We say a set is not open if there is some such that for all ,.
Example P4.9. The following sets are not open in :
and
Show solution
Consider and any . There is no such that , so the points of just to the right of are not integers, and .
Consider the endpoint of : let ; then .
Same, but with the closed endpoint: for and for .
and 5. Same.
Consider and the rational point . For every , the interval contains irrational numbers (such as , once ), which are not in ; so .
More on bases
When I say is finer than (for topologies generated by bases and ), what I mean is that for all and all containing , there is such that .
Definition P4.10. If is the collection of all open intervals in the real line, , the topology generated by is called the standard topology on the real line.
Definition P4.11. If is the basis of all half-open intervals on the reals, then the topology generated by is known as the lower limit topology. We denote this .
Definition P4.12. Let and let be the collection of open intervals along with all sets of the form . The topology generated by is called the -topology. It is denoted .
The topologies of and are strictly finer than the standard topology, but are not comparable to one another.
Definition P4.13. A subbasis for a topology on is a collection of subsets of whose union equals . The topology generated by the subbasis is defined to be the collection of all unions of finite intersections of elements of .