Mathematical Structure and Mathematical Space Hierarchies

  • #1
Hi.
I am trying to understand the images that I have posted below.

space-hierarchy_zpsuchrw0k0.png


structure-hierarchy_zpszgf98yie.png

Each layer of Mathematical Structure Hierarchy in the image in this post and Mathematical Space Hierarchy in the image in this post are: statements.
1. What do these statements of each layer of these hierarchies in the images in this post lets me do?
2. Do these statements lets me build something?
3. What if there is no Mathematical Structure Hierarchy and Mathematical Space Hierarchy?
4. Why did somebody create Mathematical Structure Hierarchy and Mathematical Space Hierarchy?
5. Do these Mathematical Structure and Mathematical Space Hierarchies exist to describe something?
6. Why each layer of Mathematical Structure Hierarchy and Mathematical Space Hierarchy be in the order as they are shown in the images above? Can they be jumbled?

I have asked these questions to know what Mathematical Space Hierarchy is and what Mathematical Structure Hierarchy is.

- Trying to connect dots.

Thanks!
 

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  • #2
Stephen Tashi
Science Advisor
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Hi.
I am trying to understand the images that I have posted below.
Where did you find these diagrams?

Have you studied the definitions of the things presented in the diagrams? The concepts of "Mathematical Space Hierarchy" and "Mathematical Structure Hierarchy" are not standard concepts of mathematics. They may be concepts invented by someone in field of mathematics education in order to give an overview of mathematical topics.

If you have studied the mathematical definitions of the things presented in the diagram, you should see that each thing is a special case of the other things that contain it. To understand why that is so, you need to know the technical definitions of each of the things.

The things in the diagrams are important mathematical concepts because they have theoretical and practical uses. Historically, each of the concepts was developed without any reference to a "Mathematical Space Heirarchy" or "Mathematical Structure Heirarchy".
 
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  • #4
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They are equivalent to statements like "every inner product space is a normed vector space" and so on. Just a graphical visualization. The set of inner product spaces is a subset of the set of normed vector spaces.

If you can show something is an inner product space then you can apply every theorem for normed vector spaces, every theorem for metric spaces and every theorem for topological spaces as well.
 
  • #5
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While I agree with the first, where there are actual inclusions, I think the second is highly questionable. "algebraic structure" isn't defined without context. It should at least be "binary operation", and "Abelian" is plain wrong, if the word "group" isn't attached to it within the context given. And it is a bit of an arbitrary property in group theoretical considerations. Why not "simple" or "finite"? Abelian refers to the binary operation in question and isn't exclusively related to groups.
 

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