The following theorem is called the Jordan Curve Theorem. Definition Let \( c \) be a loop⭕️ in the plane (also known as a Jordan curve or simple closed curve). The complement of \( c \) in \( \mathbb{R}^2 \), denoted \( \mathbb{R}^2-c \), consists of a bounded domain (the interior) and an unbounded domain…

Simply connected is a fundamental concept in mathematics, particularly in topology. Topology studies how the shape of a space behaves and how it can be continuously deformed. A space is called simply connected if any loop (closed curve) within it can be continuously shrunk to a single point. A space with this property is referred…

Understanding the Concept Let’s imagine an organism🐛 that lives attached to a two-dimensional sphere \( S^2 \) (what we normally think of as a sphere🌐). In other words, to that organism, the sphere is the universe itself. Let’s call that organism \( A \). Loops on the Sphere Now, consider any loop⭕️ (a closed curve…

When casually playing an RPG (Role-Playing Game🎮️), you might not notice it, but when it is pointed out to you, you may think “That make sense💡”. Visualizing the World Map as a Torus Imagine the square world map of an RPG🗺️. First, the top and bottom edges of the map can be identified as the…

A topic related to infinity♾️. What does \( 0.9999… \) (with an infinite number of \(9\)s after the decimal point) equal? The answer is \( 0.9999… = 1 \). The proof can be easily shown by using an infinite geometric series. Let the first term be \( a = 9/10 \), the common ratio be…