In the text below, will always refer to a metric space.
Definition: neighbourhood of a point
Let . By a neighbourhood of of radius we mean the set .
Definition: interior point
Let be a nonempty subset of . A point is said to be an interior point of if and only if there exists a neighbourhood of such that .
Definition: open set
A nonempty subset of is said to be open if and only if every point of is an interior point of .
The property of being open is related to the property of being closed by the following theorem.
Theorem: relation between open and closed sets
A subset of is open if and only if its complement is a closed subset of .
Proof. First suppose that is closed. We want to show that this implies that is open. Choose . Then is not a limit point of (if it was, then would be an element of , by definition of being closed, which is absurd, since ), and hence there exists a neighbourhood of such that is empty. But then so that is an interior point of . Hence is open.
Now suppose that is open. We want to show that this implies that is closed. Let be a limit point of . If no such exists, then contains all its limit points, and the proof is complete. If not, then every neighbourhood of is such that is not empty. But then is not an interior point of . Since is open, we must have . But then is closed, and the proof is complete.
- Rudin, Walter: Principles of Mathematical Analysis, 3rd edition, McGraw Hill, 1976.