Space

A set of objects is a collection. When you add structure to the set, you now have space.

For example:

Metric space for example introduces distance namely, $d(x,y)$.

Structure

Two types of structure we can apply is algebraic structure and toplogical structure. New rules generate new spaces.

                         ┌──────────────────────────────┐
                         │          SETS                │
                         │  (just collections of things)│
                         └─────────────┬────────────────┘
                                       │
                    ┌──────────────────┴──────────────────┐
                    │                                     │
           ALGEBRAIC STRUCTURE                    TOPOLOGICAL STRUCTURE
                    │                                     │
        ┌───────────┴───────────┐                ┌────────┴────────┐
        │                       │                │                 │
   Groups / Rings          Vector Spaces   Topological Spaces   Measure Spaces
                                                   │                │
                                                   │                │
                                           ┌───────┴───────┐        │
                                           │               │        │
                                      Metric Spaces    Uniform Spaces
                                           │
                                    ┌──────┴──────┐
                                    │             │
                              Normed Vector   Riemannian /
                                  Spaces        Metric Geometry
                                    │
                           ┌────────┴────────┐
                           │                 │
                     Banach Spaces      Inner Product Spaces
                           │                 │
                           └────────┬────────┘
                                    │
                              Hilbert Spaces