Is this one connected curve, or two? Bet you can't explain why...

Go to https://brilliant.org/Morphocular to get started learning STEM for free. The first 200 people get 20% off an annual premium subscription. One of the most fundamental properties of geometry is connectedness: when a shape is a single continuous entity. But how do you define this idea precisely so that you can apply it even to extremely bizarre shapes in very strange spaces? =Chapters= 0:00 - The Topologist's Sine Curve 1:54 - The goal of this video 3:06 - Path-Connectedness 6:56 - A new definition? 7:37 - Topology basics 10:27 - A snag 12:04 - Connectedness 2.0 and the Topologist's Sine Curve 15:48 - Who's right? 17:38 - The Ordered Square 19:32 - The connection between the definitions 21:18 - The underappreciated art of crafting definitions 22:16 - Brilliant ad =============================== CREDITS The songs used in this video are (in order of first appearance) "Dream Escape", "Checkmate", "Rubix Cube", and "Twinkle in the Night". The song "Rubix Cube" comes courtesy of Audionautix.com =============================== Want to support future videos? Become a patron at https://www.patreon.com/morphocular Thank you for your support! =============================== The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here: https://github.com/morpho-matters/morpholib