A set is essentially any collection of elements. We could define them via some arbitrary property, yet these can lead to contradictions. Hence, we use a class of axioms (ZFC axioms plus axiom of choice) to ensure logical consistency across sets and their properties. You choose elements of a set from existing consistent sets using your chosen property, rather than deduct axiomatically from scratch1.
Okay yet we just have a collection of is, xs or ys just sitting there doing nothing. What if we chose some rules to organise them? Establish for instance how close each member is, or how they relate to one another (continuity, convergence, etc)? Make our task simple, break down a non-empty set X into a set of subsets τ forming X. What rules might we want? Well first, we might want to make the set self-contained:
The empty set is contained in τ and X is in τ.
All unions of τ's subsets are also in τ.
All intersections of a finite number of τ's subsets are in τ.
We will call such a set a topology, or more accurately a topological space. For instance, all the geometry you learnt at school is in topological space. Not all sets are topologies, but all topologies are sets.
In particular, a topology carries a structure: a set of particular rules on how to get to one element in the set to another that make the set invariant to any scaling or rotation or other transformation you could think of. In particular, there exists one particularly important form of invariance. If you have two structures, and by “flipping” (more formally reverse a structure via an inverse mapping) one structure you get to the other, the two structures are the same. This invariance is called an isomorphism, and it is a gold standard in mathematics to define all sets up to isomorphism. If we can prove a set is isomorphic, then it yields an invariant structure. A set with a structure is called a space, yet not all spaces are topologies.
I almost forgot! I left out one important ingredient. Distance! Not all topologies have it, yet many do. Topologies having distance are called metric spaces. X is non-empty, with distance, a function d:X×X→R where for all members x,y,z:
d(x,y)>=0
d(x,y)=0 iff x=y
d(x,y)=d(y,x) (symmetry)
d(x,z)<=d(x,y)+d(y,z) (triangle inequality, c<=a+b for c-hypotenuse)
We can also organise sets by their operations instead. One operation is addition. Suppose that addition across all members of the set satisfies the following for all x,y,z in set X:
If x and y are in set X, x+y is in X. X is closed.
(x+y)+z = x+(y+z). X is associative.
x+y=y+x. X is commutative.
There exists an element e in X such that x+e = x. The identity of X is e.
There exists an element i in X such that x+i = e. The inverse of X is i.
If X satisfies these conditions, then X is an algebraic space. All topologies are algebraic spaces with algebraic structure (rules consistent with the above).
The operation satisfying the above need not be addition for X to be algebraic, but could be multiplication. These are binary operations: combining two elements to get to a third. If both addition and multiplication for elements in X satisfy those conditions, then X is a field. Fields are also ordered, there exists a transitive (if a>b and b>c then a>c) ordering/rank of all elements in X. A group is a structure with elements and an operation satisfying 1-4, yet all operations within a set must satisfy 1-4 for an algebraic set to be a group. The integers form an algebraic set, and are a group under addition yet not with multiplication (as the inverse operation, division, means the set is not closed).
Now we have our definitions, let us consider what a geometric space is. One might think a geometric space is a metric space with algebraic structure. Most of what we encountered in high school fits this framework. Yet not all topologies are metric spaces.
Most of us are familiar with a conventional coordinate chart with x and y axes2, or i and R^2. If all points, lines, congruence relations, and orderings within satisfy Hilbert's axioms, then we can say such a space is Euclidean:
We can see this is metric. It is also an affine space. Our choice of centre is arbitrary, so all our operations within are invariant to such. Vector spaces like R^n do not possess this property, where a properly defined origin is required to construct a basis.
Suppose that a set is Euclidean in R^2. In two-dimensions, the object is flat and unbounded. Yet such a set can be finite and we see why imminently. Now suppose we zoom out into R^3 and suddenly the set is no longer flat. The surface is curved and continuous. We have a smooth manifold: locally Euclidean, and globally not. If the set is a vector space, it is also a Riemannian manifold, so all coordinates, m or vectors or angles on it are defined in relation to their origin by an orthonormal basis. Flat earthers may stop reading now!
In general relativity, space-time is non-Euclidean, with gravity setting a curvature. Yet zoom in to a sufficiently small picture of it, and it looks like any standard graph of points, axes, and coordinates. Common examples of manifolds include S^2 = spheres, and T^2 = tori (doughnut-like). Saddles too.
A key property uniting manifolds is that they can be defined in such a manner that is invariant to your choice of dimension. We have some broad intuitive conception of size: length, area, or volume. In their respective dimensions, they represent Lebesque measures. However, such varies according to dimension. You can construct a measure of a set, like a manifold however, that is constant across dimensions, called a Haussdorf measure. Every point p on a smooth manifold also lies in a tangent space TpM, which can be a plane (as for a sphere) or an entire vector space in itself.
For any smooth manifold in R^n, its Haussdorf measure yields an index n = no of dimensions, and a constant scaled to the power of n. For instance, the area of a circle is πr^2. The volume of a sphere is (4/3)πr^3. By integrating over 2^2 times area, you see the power is equal to the number of dimensions. For area, 2=n for R^2 too, and we get to it by integrating the circumference w.r.t. r. We integrate over πr scaled by a constant 2^n to get area then volume and so on, and this constant is the Haussdorf measure in this case.
Now let us work with the interval in R, [0,1]. We remove the middle third repeatedly, until we get to such a set with zero length and magnitude zero yet an uncountable number of points. The set is too large to have no dimension, yet too small to be in R. This is an example of a fractal. Yet with these measures, you simply just define some concept of size or cardinality in a manner satisfying invariance, which does not adequately distinguish algebraic from geometric spaces.
Moreover, there are also examples of manifolds that are algebraic spaces without distances, so my first definition of a geometric space yields a contradiction. Suppose for a point on a smooth manifold, we take two vectors lying in that tangent plane. We can get a function ωp:TpM×TpM→R, which is the 2-form of the manifold. By combining these two vectors via multiplication, we get one real number.
Now suppose such a function is linear, differentiable, yet antisymmetric, and also closed. ωp(u,v) = -ωp(v,u) for vectors u,v in TpM. For u=/=0, if ω(u,v)=/=0 then the 2-form is nondegenerate. You will never reach zero from a nonzero vector with an operation combining another vector under this property. With nondegeneracy, linearity, antisymmetry, a closed 2-form is symplectic. Symplectic manifolds are manifolds with symplectic form, and without symmetry they cannot be metric spaces. Yet we still consider them geometries by any reasonable definition. Curves, lines, and planes!
Okay, so perhaps we should just eliminate the word geometry from the dictionary in favour of topology. Get a smooth, hence differentiable manifold M, and add metric structure to it. Here, we start with the topology, then add additional structure to form an algebraic space.
However, we can also get a topology from an algebraic space. Let V be a vector space, with an inner product VxV→R. That inner product, normalised, gives a metric d(x,y) = ||x-y||. That metric yields a set of points with an empty set subset (d(x,x) = 0), its subsets and unions of subsets also distances, and intersections of subsets of differences being distances. We therefore get a topology via an algebraic structure, so some algebraic structures themselves can be sufficient for a topology and necessarily generate a topology. You get the same results via rings.
The standard answer is to simply define sets in Euclidean space by isomorphism (as we discussed), and also topologies by homeomorphism. Both produce invariance in structure if such structure is reversed.
Yet structure alone does not establish a clear definition of geometry. The distinction between algebraic vs geometric spaces is blurred. Not all topologies nor metrics are algebraic. Not all algebraic spaces are topologies. Will superintelligence finally resolve the question as to what a geometry actually is?
In that sense, with completeness and independence, you have a solution bridging Hilbert's formalism vs Brouwer's constructivism. You can construct sets from arbitrary chosen axioms, so long as such axioms are logically consistent with other proven facts. More formally, we consider defining sets up to isomorphism to be the gold standard, as this is sufficient (albeit not necessary) to demonstrate logical consistency. This also puts to bed any notion of epistemology other than Cartesian rationalism (and note that inference is probabalistic, and probability itself operates according to formal rules).
These of course form a polynomial ring for the complex number set. Isomorphically, they are an equivalent metric space. Their points are automorphic (swap i for y=1 etc, complex and Cartesian coordinates are just different notation styles), and distance is preserved between them too. They are not algebraically isomorphic. Vector multiplication and that operation in the complex set behave differently. We can also see, if we make generate polynomials in (x,y) in a finite manner, that such a ring is also Noetherian, so by Hilbert's basis theorem, any polynomial function can get to any point in R^2 with real coefficients. You can visualise this clearly in complex space: every complex number is of the form a+bi where a,b are real. I digress, but seeing how complex vs Euclidean R^2 spaces differ and share similarities is exactly the sort of thing mathematicians do. We teach years of the boring crap before going into the engaging stuff.