To every set, we may associate its boundary, its interior and its closure.
Let be a topological space and a subset. The boundary of is defined as the set
Proposition (a set is closed iff it contains its boundary):
Let be a topological space. A set is closed if and only if .
Proof: Suppose first that contains its boundary. We show that is open. Indeed, let . Note that a set is open if and only if it contains a neighbourhood of each of its points. If there does not exist a neighbourhood so that and , then since , we get , a contradiction. Suppose then that is closed, and let . If there does not exist a neighbourhood of contained in , then , contradiction.
Let be a topological space and be a set. The interior of is defined to be
Proposition (the interior of a set is open):
Let be a topological space and . Then is open.
Proof: Let . Then there exists a neighbourhood of contained in , for otherwise, all neighbourhoods of intersect (since ) as well as , since if is a neighbourhood that does not intersect , but that does intersect we may pass to an open subset of which is still a neighbourhood of and intersects (otherwise we are done), so that , so that is a neighbourhood of , so that intersects . Hence is open by the characterisation of topologies by neighbourhood systems.
Let be a topological space and let . The closure of is defined to be
Proposition (the closure of a set is closed):
Let be a topological space and a set. Then is closed.
Proof: We show that is open. Let . Suppose that does not contain a neighbourhood of . Then every neighbourhood of intersects either or . If is a neighbourhood of that intersects but not , we may pick to be open by passing to a subset of , and then we'll still have some point , since otherwise the original would intersect . But implies by definition that all neighbourhoods of (e.g. ...) intersect , so that we obtain a contradiction, so that all neighbourhoods of intersect and , contradiction.
Proposition (the closure of a set equals the intersection of all closed subsets contained within it):
Let be a topological space and a set. Then
Proof: We have seen that the closure of a set is closed, so that the right hand side is contained in the left hand side. Further, let and let be a closed superset of . Suppose that , then there exists so that since is open. But then neither nor , since , a contradiction.
Let be a topological space and a subset. is called dense in iff .
Proposition (characterisation of density):
Let be a topological space and . The following are equivalent:
- is dense in
- for all nonempty open subsets , we have
- the interior of is empty
Proof: The equivalence of 1. and 3. follows because of the fact that , which in turn can be seen from the definition of interior and closure and the fact that in general. Now we prove the equivalence of 1. and 2. Indeed, suppose first that is dense in , and let be open and nonempty. If , then is a proper closed superset of , whence , a contradiction. Suppose now that 2. holds, and let be any closed superset of . Then suppose that for some . Then is a nonempty open set that does not intersect , contradicting 2. Hence, , and we conclude since the closure of a set equals the intersection of all closed subsets contained within it.
Definition (greatest lower bound topology):
Let be a set and let be a family of topologies on . The topology given by
being the largest topology that is contained in all the , is called the greatest lower bound topology
Definition (topology generated by a set):
Let be a set and let . The topology generated by is the topology given by
Note that these two are topologies since the intersection of topologies is again a topology.
Proposition (characterisation of the generated topology):
Let be a set and . Let be the topology generated by . Then, upon defining
Proof: Clearly, , since is a topology that contains all the . On the other hand, we show that is a topology, thereby proving that by definition of the topology generated by a set. Indeed, note first that upon choosing and , the empty intersection. Then note that by choosing . Suppose then that and write, for ,
- , and , .
Finally, suppose that for we have , and write
- , being a finite intersection of sets of .
Thus, we've shown that is a topology.
Definition (least upper bound topology):
Let be a set, and let be topologies on . The least upper bound topology on is defined to be the topology generated by
Proposition (least upper bound topology is smallest topology containing the given topologies):
Let be a set, and let be topologies on . Let be the least upper bound topology on defined by the . Then is minimal among all topologies that contain all ().
Proof: By definition of the topology generated by a set, any topology that contains all the topologies will contain .
Proposition (topologies on a fixed set are a complete lattice):
Let be a set, and let be a family of topologies on . Then upon ordering by inclusion, there exists a least upper bound of , as well as a greatest lower bound. Thus, the topologies on a set, when ordered by inclusion, form a complete lattice.
Proof: We have already seen that there exists a least upper bound topology. Then note that topologies form an algebraic variety, where union and intersection are arbitr-ary operations, whereas the whole set and the empty set are -ary operations. Then the existence of the greatest lower bound (and that it equals the intersection of the topologies in question) follows immediately since the greatest lower bound structure is the intersection.
Final and initial topologies[edit | edit source]
Proposition (preimage of a topology is a topology):
Let be a function from a set to another set , and let be a topology on . Then is a topology on .
Proof: This follows instantly from the formulae
- and for any family of subsets of
Definition (initial topology):
Let be a set and let be a family of topological spaces. Let further () be functions. The initial topology on is the least topology that contains all the topologies .
Definition (product topology):
Let be a family of topological spaces. Define the set
- , the Cartesian product.
The product topology is a topology on , namely the initial topology with respect to the collection of all projections
- , .