dwsmith said:Explain why all the end points of the closed intervals that comprise $C_n$ are in the Cantor set $C_n$.
I understand why this is true but I don't know how to explain it.
The Cantor ternary set, also known as the Cantor set, is a fractal that is constructed by repeatedly removing the middle third of a line segment. It is named after Georg Cantor, a mathematician who first described the set in 1883.
The Cantor ternary set is constructed by starting with a line segment and dividing it into three equal parts. The middle third is then removed, leaving two line segments. This process is then repeated on each of the remaining line segments, creating smaller and smaller line segments. The final set is the union of all the remaining line segments after infinite iterations.
The Cantor ternary set is a self-similar fractal, meaning that it is made up of smaller copies of itself. It has a fractal dimension of ln(2)/ln(3), which is approximately 0.6309. It is also uncountable, meaning that it has an infinite number of points, despite being a one-dimensional set.
The Cantor ternary set has significant implications in mathematics, as it was one of the first examples of a set with a non-integer dimension. It also has applications in other fields, such as computer science and physics, and has been used to model natural phenomena such as coastlines and snowflakes.
The Cantor ternary set is related to other fractals through its self-similarity and non-integer dimension. It is a subset of the Sierpinski triangle and can be created by repeatedly removing the middle third of each triangle in the Sierpinski triangle. It is also a special case of the Cantor dust, which is a similar fractal constructed by removing the middle third of a square instead of a line segment.