Fluffman4 said:We defined the closure of a subset A of a topological space as the intersection of all closed sets containing A. We basically want to find the smallest closed set containing A for closure.
The closure of the rational numbers refers to the set of all real numbers that can be reached by taking limit points of the rational numbers. This includes the rational numbers themselves, as well as any irrational numbers that can be approached by a sequence of rational numbers.
In standard topology, the closure of a set is defined as the union of the set and all of its limit points. So, in the case of the rational numbers, the closure would be the set of all rational numbers and any real numbers that can be approached by a sequence of rational numbers.
The closure of the rational numbers is important because it helps to fill in the gaps between rational numbers and allows us to have a complete understanding of the real numbers. It also helps in the study of continuity and convergence in analysis.
No, the closure of the rational numbers is not the same as the closure of the integers. While the closure of the rational numbers includes both the rational and irrational numbers, the closure of the integers only includes the integers themselves.
The Cantor set is a perfect example of a subset of the real numbers that is uncountable and has a measure of zero. It can be shown that the Cantor set is contained within the closure of the rational numbers, as it includes all points that can be approached by a sequence of rational numbers. Therefore, the closure of the rational numbers is a larger set that contains the Cantor set.