Meaning of the word topological

In summary, the term "topological action" refers to a transformation that is continuous, resulting in objects being topologically equivalent before and after the transformation. This can be further explained by the concept of topological equivalence, which can be found in the field of topology. The term "topological" may also be associated with actions being continuous or discontinuous. Additionally, topological quantum field theory, a concept studied by Ed Witten, may also relate to this topic.
  • #1
jinbaw
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when we say "a topological action", do we only mean that the action is metric free? or is there some other meaning for this expression? What does the word topological mean exactly? Thanks!
 
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  • #2
It usually means that the given transformation is a continuous one so that the object before the transformation and the object after the transformation are 'topologically' equivalent. If you familiar with the concept of topological equivalent, it may be helpful.
 
  • #3
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1. What is the meaning of the word "topological"?

The word "topological" refers to the study of mathematical spaces and how they are related to each other through deformations, without considering their specific measurements or geometric properties. It is a branch of mathematics that focuses on the study of the properties that remain unchanged when a space is stretched or twisted.

2. How is topology different from geometry?

Topology and geometry are both branches of mathematics that deal with the study of shapes and spaces. However, topology focuses on the properties of shapes that remain unchanged under continuous deformations, while geometry considers the exact measurements and properties of shapes.

3. What are some real-world applications of topology?

Topology has various applications in different fields such as physics, computer science, and engineering. It is used in the study of fluid dynamics, material science, and the design of computer networks. Topological concepts are also applied in the analysis of brain networks, DNA structures, and the study of weather patterns.

4. How is topology used in data analysis?

Topological data analysis is a method that uses topological concepts to analyze and understand complex data sets. It can be used to identify patterns and relationships in data that may not be apparent through traditional methods. This approach has been applied in fields such as biology, finance, and social sciences to gain insights from large and complex data sets.

5. What are some key concepts in topology?

Some key concepts in topology include continuity, connectedness, compactness, and homotopy. Continuity refers to the idea that small changes in a space do not affect its properties. Connectedness is the property of a space that allows you to move from any point to another without breaking the space. Compactness refers to the idea that a space is finite and has a boundary. Homotopy is the concept of deforming one shape into another without tearing or gluing.

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