This post concerns sets of real numbers. It is based on the book Introduction to Topology, exercise 5, page 9, of Menselson [1]. It aims to prove some properties of the intersection and union of sets of real numbers.
Let be a set. We denote by an indexed family of subsets of . The union of the subsets is noted . It represents the set of all elements in that belongs to at least one subset . The intersection of the subsets is noted . It represents the set of all elements in that belongs to every subset .
Here, we will focus on indexed families of subsets of positive real numbers to manipulate families of sets. The proofs will involve classical techniques: double inclusion and contradiction.
Let be the set of real numbers that are greater than . Note and .
- The intersection of open sets of positive real numbers is empty,
The proof is by contradiction. Assume that . It exists such that . By definition of , . By definition of the intersection of sets, , which contradicts the assumption. So, .
- The union of open sets of positive real numbers is the set of positive real numbers,
The proof is by double inclusion. Let . By definition of the union, it exists such that . As , then . So . For the other way, let . It exists such that . So, . By definition of the union, . So . Finally, .
- The intersection of closed sets of positive real numbers is the singleton ,
The proof is by double inclusion. For all . It implies that and . For the other way, assume that it exists such that . By definition, and by definition of the intersection, . So, there is no element of in and . Finally, .
- The union of closed sets of positive real numbers is the union of the set of positive real numbers and the singleton ,
The proof is by double inclusion. Let . By definition of the union, it exists . By the definition of a subset, or . And by definition of the union, . For the other way, let . If , then by definition, . It exists such that . It implies that . By definition of the union, and . Finally, .
References:
[1] Mendelson, B., 2012. Introduction to Topology: Third Edition. Courier Corporation.