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

In summary, the conversation discusses the challenge of understanding a topology of a space when it is defined indirectly through convergence. The questions raised are how to interpret this type of topology and how it relates to the structure of open sets.
  • #1
wayneckm
68
0
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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  • #2
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.
 

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

What is topology?

Topology is a branch of mathematics that studies the properties of geometric figures and spaces that are preserved under continuous deformations, such as stretching, bending, and twisting, but not tearing or gluing.

What is convergence in topology?

In topology, convergence refers to the idea of a sequence of points or objects in a space approaching a limit point or object, where the distance between the points or objects and the limit point or object becomes smaller and smaller.

What is the difference between sequential convergence and topological convergence?

Sequential convergence is a type of convergence where a sequence of points or objects in a space approaches a limit point or object, while topological convergence is a more general type of convergence where the distance between points or objects and a limit point or object becomes smaller and smaller, but not necessarily in a sequence.

What is the importance of topology and convergence in mathematics?

Topology and convergence are important concepts in mathematics because they allow us to study the properties of spaces and objects in a flexible and abstract way. They are used in a wide range of fields, including geometry, analysis, and physics.

What are some applications of topology and convergence in real-world problems?

Topology and convergence have many applications in real-world problems, such as in computer science, where they are used to design efficient algorithms and data structures, and in economics, where they are used to study market equilibrium and optimization problems.

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