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Not surprizingly, when he says "no segment", he is not talking about any segment!julypraise said:
Again, you are misunderstanding. He is not saying that 3^-m< (beta- alpha/6, he says "If" 3^-m< (beta- alpha)/6. That is an hypothesis.
The Cantor set is a mathematical set that is constructed by repeatedly removing the middle third of a line segment. The resulting set is a perfect example of a fractal, as it has a self-similar structure at different scales. It was first described by German mathematician Georg Cantor in 1883.
The Cantor set is often mentioned in the famous textbook "Principles of Mathematical Analysis" by Walter Rudin, also known as "Baby Rudin". In this book, the Cantor set is used as an example to introduce the concept of a perfect set, which is a set that is closed, contains no isolated points, and has a cardinality of the continuum.
The Cantor set is important in mathematics because it serves as a fundamental example of a set with interesting properties such as being uncountable, having a measure of zero, and being nowhere dense. It also has connections to other areas of mathematics, such as fractal geometry and topology.
Yes, the Cantor set is considered a fractal due to its self-similar structure at different scales. This means that as you zoom in on different parts of the set, you will see similar patterns repeating themselves. Fractals have applications in various fields, including computer graphics, biology, and physics.
The Cantor set is constructed by starting with a line segment and repeatedly removing the middle third of each remaining line segment. This process is continued infinitely, resulting in a set that contains infinitely many points but has a measure of zero. The construction of the Cantor set can also be generalized to higher dimensions.
