Interior of a Set

Ifis a subset of a setthen the interior of the setis writtenand is equal towhereis the closure ofWe write

We can also think of the interior of a set in terms of open sets. The interior of a setis the largest open set containing A. We can write

Example: Supposewith the usual metricon A has no interior since there are no open sets inwith the usual metric, so

Proof

Suppose

andare equivalent statements. If D subset A then X-A subset X-D=C. Since is open, C is closed.

We obtain