Real Analysis - Open and Closed Sets

Question

U = {(x, y) ∈ R n : x 2 + 4y 2 < 4}

A = {(x, y) : x ∈ Q; y 2 ≥ x}

B = {(x, y) : x /∈ Q; y 2 ≥ x}

Show that if U is open and A is closed, then U \ A = {x ∈ U : x /∈ A} is open.

What can be said about A \ U?

Give reasons.

Answer 1

Part 1: Show that U∖AU \setminus AU∖A is open

We are given:

1. U is open
2. A is closed

Recall:

U∖A={x∈U:x∉A}

This can be rewritten as:

U∖A=U∩A^cU

where A^c is the complement of A

Key facts:

1. If A is closed, then A^c is open.
2. The intersection of two open sets is open.

Since:

1. U is open
2. A^c is open

then:

U∩A^c is open.

Conclusion:

U∖A is open

Part 2: What can be said about A∖U ?

Now consider:

A∖U = A∩U^c

We know:

1. A is closed
2. U is open ⇒ U^c is closed

The intersection of two closed sets is closed.

Thus:

A\ U is closed​

Final Summary

If:

1. U is open
2. A is closed

Then:

U∖A is open

A∖U is closed

This follows directly from:

1. complements of closed sets are open,
2. complements of open sets are closed,
3. intersections preserve openness and closedness.