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An open subset of a metric space is a set that contains only interior points.

In the text below, (X, d) will always refer to a metric space.

Definition: neighbourhood of a point

Let x \in X. By a neighbourhood of x of radius 0 < r \in \mathbf R we mean the set N_r(x) = \{ y \in X | d(x, y) < r \}.


Definition: interior point

Let E be a nonempty subset of X. A point x \in E is said to be an interior point of E if and only if there exists a neighbourhood N of x such that N \subset E.


Definition: open set

A nonempty subset E of X is said to be open if and only if every point of E is an interior point of E.


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 E of X is open if and only if its complement E^c is a closed subset of X.

Proof. First suppose that E^c is closed. We want to show that this implies that E is open. Choose x \in E. Then x is not a limit point of E^c (if it was, then x would be an element of E^c, by definition of being closed, which is absurd, since x \in E), and hence there exists a neighbourhood N of x such that N \cap E^c is empty. But then N \subset E so that x is an interior point of E. Hence E is open.

Now suppose that E is open. We want to show that this implies that E^c is closed. Let x be a limit point of E^c. If no such x exists, then E^c contains all its limit points, and the proof is complete. If not, then every neighbourhood N of x is such that N \cap E^c is not empty. But then x is not an interior point of E. Since E is open, we must have x \in E^c. But then E^c is closed, and the proof is complete.

QED.


References

  • Rudin, Walter: Principles of Mathematical Analysis, 3rd edition, McGraw Hill, 1976.

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