What is the relationship between topology and convergence in defining open sets?

[wayneckm]

Hello all,


Sometimes I come across the situation that a topology of a space is defined indirectly through some convergence mode. I can understand when we are given a topology, we can define the convergence of a sequence w.r.t this topology. However, if we start with saying the space is endowed with topology of convergence in some sense, apparently it is quite hard to imagine the structure of a open set in such space. So I have the following questions:

1) When we come across this kind of topology (by convergence mode), how should we understand it?

2) How can this convergence "imply" the structure of a open set?

Thanks.


Wayne

 

[arkajad]

For instance you can define a closed set to be a set that contains all its limit points and then define an open set as the complement of a closed set.