Most pure mathematics takes as a starting point a set of objects. If things stopped there, we would be dealing just with set theory; but we branch into other areas of pure mathematics by adding structure to the set.
Defining such a structure may involve calling out particular elements of the set or particular subsets whose element have some particular relationship to one another; or it may involve some mapping from one element in the set to other or an operation that takes two elements of the set and produces a third.
Calling out particular subsets is the basis for the branch known as topology. If the choice of subsets meets certain criteria (which we'll get to shortly), the set (along with the called-out subsets) is called a topological space.
Defining an operation that takes two elements of the set and produces a third that is also in the set (think of adding two numbers or concatenating two strings) is the basis for the branch known as group theory. If the operation meets certain criteria (which we'll also get to shortly), the set and the operation is called a group.
Some structures involve reference to one or more additional sets (such as the set of real numbers). For example, one might define an operation that takes two elements of a set and gives a number that can be thought of as the "distance" between the two elements. As long as that operation follows certain rules (such as the distance between two distinct elements always being positive and the distance between an element and itself always being 0) then the operation is called a metric and the set and the operation is called a metric space.
In the next few entries in this project, we'll take a look at the criteria necessary for a set + operation to be a group and for a set + collection of subsets to be a topological space.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.