In a nutshell, a mathematical space is a set of objects equipped with a structure that defines
how its objects are allowed to interact.
Super vague description right? But that's actually the reason why spaces are so powerful.
For example, to model real life as a mathematical space, the set could be {Matter, Antimatter, Photons, ...}
and the structure could be a list of the laws of physics.
Let's say you found a DVD copy of this "real-life space" deep in the pyramids of Giza.
That DVD would be the most valuable item ever. it would contain every bit of information
in the universe and every law of physics.
With it, mathematicians could simulate a perfectly accurate model of real life allowing us
to exactly predict the future and perfectly replay the past.
Even better; With a sufficiently powerful VR setup, we could insert people into simulations
of anywhere and anytime in the universe.
Again, this isn't even close to possible. As you will see, the current study of mathematical
spaces is unmeasurably small compared to the real-life space.
For the spaces we have created, there is no set method for making a new structure.
If a brilliant mathematician decides to manipulate objects with a new set of rules,
then they've discovered a new space.
A quick historical example: Euclid created a space called the Euclidean plane.
2000 years later, Descartes took the Euclidean plane and added a coordinate structure.
Even though Descartes started with a pre-existing space -- the result was a new space called the Cartesian plane.
It's important that new spaces are fundamentally diffent from their predecessors.
If there's a turn left(TL) space and a turn right(TR) space, then we have this structure-preserving map
called 'upside down' that can change between them without violating either of the spaces structures.
When this happens, we can say that the TL space and the TR space are actually the same thing.
This wasn't the case with Euclid and Descartes because the only way to transpose them is by
adding or subtracting the coordinate structure. Thus any maps between the two are
not structure-preserving.
There aren't that many spaces out there either. The truth is, modern mathematics only
has about 10 major fundamental kinds of spaces.
What do you suppose were some of the first mathematical spaces humanity discovered?
What sets were they thinking of and what rules/structures were they using.
Try to come up with a few examples of objects you think are definitely not spaces.
Let's start with a familiar space to match our intuition with the abstract.