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Convex Topology
aka Convexly Topological Spaces

Invented by Douglas Moreman

Rene Descartes (and possibly others) created Analytic Geometry by combining the ancient field of Geometry with the new Algebra of Numbers. (by 1637, with "Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences")
Before 1850, Hermann Grassmann created an algebra for geometric points, creating what is now known as "Linear Algebra" -- which I prefer to think of as "Algebraic Geometry."

Even more recently, abstract Algebra has been joined with Topology in "Algebraic Topology."
There is a subject called "Topology" and there is a subject called "Convexity."
Topology generalizes the concept of a limit of a set of points and convexity generalizes the concept of "between."
Convex Topology can be thought of a combining of "Topology" and "Convexity" and
can also be thought of as an abstracting from "Normed Linear Space" by replacing linear structures with convex sets and replacing "distance" (from one point to another) with a weaker way to define limits.

Here begin the actual foundations of Convex Topology