topology via converging nets
Given a topological space , one can define the concept of convergence of a sequence, and more generally, the convergence of a net. Conversely, given a set , a class of nets, and a suitable definition of “convergence” of a net, we can topologize . The procedure is done as follows:
Let be the class of all pairs of the form where is a net in and is an element of . For any subset of with , we say that a net converges to with respect to if is eventually in . We denote this by . Let
Then is a topology on .
Proof.Clearly for any pair . In addition, is vacuously true. For any , we want to show that . Since is eventually in and , there are (where is the domain of ), such that and for all and . Since is directed, there is a such that and . It is clear that and that any we have that as well. Next, if are sets in , we want to show their union is also in . If is a point in then is a point in some . Since with is eventually in , we have that is eventually in as well. ∎
Remark. The above can be generalized. In fact, if the class of pairs satisfies some “axioms” that are commonly found as properties of convergence, then can be topologized. Specifically, let be a set and again be the class of all pairs as described above. A subclass of is called a convergence class if the following conditions are satisfied
is a constant net with value , then
implies for any subnet of
if every subnet of a net has a subnet with , then
suppose with , and for each , we have that , with . Then , where is the net whose domain is with , given by .
If , we write or . The last condition can then be visualized as
which is reminiscent of Cantor’s diagonal argument.
Now, for any subset of , we define to be the subset of consisting of all points such that there is a net in with . It can be shown that is a closure operator, which induces a topology on . Furthermore, under this induced topology, the notion of converging nets (as defined by the topology) is exactly the same as the notion of convergence described by the convergence class .
In addition, it may be shown that there is a one-to-one correspondence between the topologies and the convergence classes on the set . The correspondence is order reversing in the sense that if as convergent classes, then as topologies.
|Title||topology via converging nets|
|Date of creation||2013-03-22 17:14:27|
|Last modified on||2013-03-22 17:14:27|
|Last modified by||CWoo (3771)|