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

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The discussion centers on the relationship between topology and convergence in defining open sets. It highlights that a topology can be defined indirectly through convergence modes, which complicates the understanding of open sets. Wayne clarifies that a closed set is defined as one containing all its limit points, leading to the definition of open sets as their complements. This establishes a clear connection between convergence and the structure of open sets in topological spaces.

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wayneckm
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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
 
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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.
 

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