High Level Picture of Space and Algebra

Here is the space hierarchy:

+----------------------+
| Metric Space         |
+----------------------+
        |
        V
+----------------------+
| Normed Space         |
+----------------------+
        |
        V
+----------------------+
| Banach Space         |
+----------------------+
        |
        V
+----------------------+
| Inner Product Space  |
+----------------------+
        |
        V
+----------------------+
| Hilbert Space        |
+----------------------+
        |
        |
        |                 +------------------------+
        |                 | Sobolev Spaces         |
        |                 +------------------------+
        |                         |
        V                         V
+----------------------+   +------------------------+
| Euclidean Space      |   | $L^p$ Spaces       |
+----------------------+   +------------------------+
                                     |
                                     V
                      +------------------------+
                      | Lebesgue Spaces        |
                      +------------------------+

Additional branches:

Topological Space
|
V
+—————–+
| Locally Compact |
| Hausdorff Space |
+—————–+
|
V
+—————–+
| Topological |
| Vector Space |
+—————–+
|
V
+——————-+
| Banach Manifold |
+——————-+

+------------------+
| Algebraic Structure |
+------------------+
        |
        V
+--------------------+
| Ring               |
+--------------------+
        |
        |
        +-------------------+
        |                   |
        V                   V
+-----------------+   +------------------------+
| Commutative     |   | Non-Commutative Ring   |
| Ring            |   |                        |
+-----------------+   +------------------------+
        |
        V
+--------------------+
| Integral Domain    |
+--------------------+
        |
        V
+--------------------+
| Field              |
+--------------------+
        |
        |
+-----------------------------------+
| Subfield / Extension Field        |
+-----------------------------------+

Additional Structure for Modules:

+------------------+
| Module           |
+------------------+
        |
        V
+-------------------+
| Vector Space      | (Module over a field)
+-------------------+

Algebraic Structure: The broadest category encapsulating rings, fields, and modules.
Ring: Fundamental algebraic structure supporting addition and multiplication.
- Commutative Ring: Special type where multiplication is commutative.
  - Integral Domain: Commutative ring without zero divisors.
    - Field: Commutative ring with multiplicative inverses for all non-zero elements.
      - Subfield/Extension Field: More specific or extended structures within fields.
- Non-Commutative Ring: Rings without commutative multiplication.
Module: Generalization of vector spaces where scalars form a ring.
- Vector Space: Special case of a module where scalars form a field.

Additional notes on conjugacy classes (not groups) and cosets and normal subgroup, using S3 as example to explain: Conjugacy classes group elements that are similar in the structure.Cosets are the group split into parts based on a subgroup.A normal subgroup is a special subgroup where the structure is preserved even when elements are conjugated.