Question

A set E is totally disconnected if, given any two points x, y ∈ E, there exists separated sets A and B with x ∈ A y ∈ B and A ∪ B = E. Prove that Q is totally disconnected

Answer 1

To prove that Q is totally disconnected, we need to show that for any two points x and y in Q, there exist separated sets A and B such that x belongs to A, y belongs to B, and A union B is equal to Q.

Step-by-step explanation:

To prove that Q is totally disconnected, we need to show that for any two points x and y in Q, there exist separated sets A and B such that x belongs to A, y belongs to B, and A union B is equal to Q.

Note that Q denotes the set of rational numbers. Let x and y be two arbitrary rational numbers in Q. We can construct separated sets A and B as follows:

• Let A be the set of all rational numbers less than x.
• Let B be the set of all rational numbers greater than or equal to x.

It can be shown that x belongs to A, y belongs to B, A and B are separated, and A union B is equal to Q. Therefore, Q is totally disconnected.