Find two open sets A and B, such that A is subset of B, A is not equal

Thread starter math25
Start date Apr 21, 2012
Tags

Sets

In summary, an open set is a set that does not include its boundary points and has a small enough neighborhood around each point that is completely contained within the set. A set can be both open and closed, such as the set of all real numbers between 0 and 1. One way to find two open sets A and B where A is a subset of B is to choose any open set A and take a smaller open set B contained within A. It is important for A and B to be open sets because they are fundamental in topology and have various practical applications in fields such as physics and engineering. There are no other requirements for A and B besides A being a subset of B.

Apr 21, 2012

math25

Find two open sets A and B, such that A is subset of B, A is not equal to B, and m(A)=m(B)

Can I use these two sets?

A=(0,2) B=(0,1) U (1,2)

thanks

Physics news on Phys.org

Apr 21, 2012

Dick

Science Advisor

Homework Helper

Almost. Your example has B a subset of A. You want the reverse.

Apr 21, 2012

math25

you are right, thanks

1. What is the definition of an open set?

An open set is a set in a topological space that does not contain its boundary points. In other words, for every point in an open set, there exists a small enough neighborhood around that point that is completely contained within the set.

2. Can a set be both open and closed?

Yes, a set can be both open and closed. For example, in the real number line, the set of all real numbers between 0 and 1 is both open and closed.

3. How can I find two open sets A and B such that A is a subset of B?

One way to find such sets is to choose any open set A and then take a smaller open set B that is contained within A. For example, if A is the set of all real numbers between 0 and 1, then B could be the set of all real numbers between 0 and 0.5, which is a subset of A.

4. Why is it important for A and B to be open sets?

It is important for A and B to be open sets because the concept of open sets is fundamental in topology and helps define many important properties and concepts in mathematics. Additionally, open sets have many practical applications in fields such as physics, engineering, and computer science.

5. Are there any other requirements for A and B besides A being a subset of B?

No, there are no other requirements for A and B besides A being a subset of B. However, it is important to note that A and B must also be open sets, as specified in the original question.