Interior (topology)

In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S.

The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.

The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty). The interior and exterior are always open while the boundary is always closed. Sets with empty interior have been called boundary sets.[1]

If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is completely contained in S. (This is illustrated in the introductory section to this article.)

This definition generalizes to any subset S of a metric space X with metric d: x is an interior point of S if there exists r > 0, such that y is in S whenever the distance d(x, y) < r.

This definition generalises to topological spaces by replacing "open ball" with "open set". Let S be a subset of a topological space X. Then x is an interior point of S if x is contained in an open subset of X which is completely contained in S. (Equivalently, x is an interior point of S if S is a neighbourhood of x.)

The interior of a subset S of a topological space X, denoted by Int S or S°, can be defined in any of the following equivalent ways:

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

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.

The above statements will remain true if all instances of the symbols/words

For more details on this matter, see interior operator below or the article Kuratowski closure axioms.

In general, the interior operator does not commute with unions. However, in a complete metric space the following result does hold:

The result above implies that every complete metric space is a Baire space.

The red shapes are not interior-disjoint with the blue Triangle. The green and the yellow shapes are interior-disjoint with the blue Triangle, but only the yellow shape is entirely disjoint from the blue Triangle.

Two shapes a and b are called interior-disjoint if the intersection of their interiors is empty. Interior-disjoint shapes may or may not intersect in their boundary.