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How Did Mathematicians Abstract the Idea of Distance?

CORRECTION: 17:26 the set Q should actually be defined as the set of all numbers that can be represented as a fraction of two integers, not as "any number with finite definition". That is Q is the set of all a/b, where a,b are integers. Slightly smaller correction: the term "circle" actually just refers to the boundary technically, so all points with distance from the origin equal to a given radius. On the other hand, a "disc" refers to all points with distance from the origin less than the radius. Similarly, the "sphere" is analogous to the "circle", referring to all 3D points with a distance from the origin equal to the radius. The "ball" is also analogous to the "disc" as the set of points in 3D space with a distance from the origin less than the radius. In this video this terminology is not distinguished, but it's useful to know that this is technically how these objects are referred to. Same with sphere (boundary) vs ball (solid). An introductory video on the topic of topology, starting with the notion of metric spaces. I hope you enjoy this video and find it useful either in your study of math, or just interesting intellectually. Please do let me know what you think of the animation as a whole, this is the longest animated video I've ever made by far! All feedback is greatly appreciated! To preview drafts of videos and give input into upcoming content, check out our Patreon: patreon.com/PiSquared Timestamps: 0:00 Intro 0:55 Defining Metric Spaces 3:33 Exploring the Interior of a Subset 6:17 Closed Subsets 8:39 Theorems of Metric Spaces 15:51 Incompleteness? 18:24 Ending