Interior (topology) - Examples

Examples

• In any space, the interior of the empty set is the empty set.
• In any space X, if, int(A) is contained in A.
• If X is the Euclidean space of real numbers, then int = (0, 1).
• If X is the Euclidean space, then the interior of the set of rational numbers is empty.
• If X is the complex plane, then int
• In any Euclidean space, the interior of any finite set is the empty set.

On the set of real numbers one can put other topologies rather than the standard one.

• If, where has the lower limit topology, then int = [0, 1).
• If one considers on the topology in which every set is open, then int = .
• If one considers on the topology in which the only open sets are the empty set and itself, then int is the empty set.

These examples show that the interior of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.

• In any discrete space, since every set is open, every set is equal to its interior.
• In any indiscrete space X, since the only open sets are the empty set and X itself, we have int(X) = X and for every proper subset A of X, int(A) is the empty set.