The Cantor set is a perfect, self-similar fractal set that is constructed by repeatedly removing the middle third of a line segment. It is named after mathematician Georg Cantor who first described it in the late 19th century.
The Cantor set has significant implications in mathematics, particularly in the study of topology and measure theory. It is an example of a set that is uncountable, meaning it has an infinite number of elements, but cannot be put in one-to-one correspondence with the set of natural numbers.
This statement means that if you take the Cantor set and add it to itself, the resulting set will contain at least one open set. An open set is a set of points that does not include its boundary, and is often a fundamental concept in topology and analysis.
The proof for this statement is based on the fact that the Cantor set is a perfect set, meaning it has no isolated points. This allows for the construction of open intervals within the Cantor set, which can then be used to show that C+C contains an open set.
The statement has significant implications in the field of analysis, as it provides an example of a set that is "small" in terms of measure, but still contains an open set. This challenges our intuition about the relationship between measure and topological properties, and has led to further research and developments in these areas of mathematics.