Interior Exterior And Boundary Of A Set

Interior closure and boundary We wish to develop some basic geometric concepts in metric spaces which make precise certain intuitive ideas centered on the themes of interior and boundary of a subset of a metric space.

Interior exterior and boundary of a set. Boundary of a set A. It is a one dimensional point set while the other two are two dimensional point sets as is the plane as a whole. Complement of an open set.

The closure of Yis the set Y T AYAand Ais closed in X. The point w is an boundary point of the set A if every neighborhood of w meets both A and Ac has a non-empty intersection D w A 6 6 D. Boundaries set out in this manner and clarified with the above notations remove any confusion around ownership of the relevant building structure or the interpretation of interior as applying to the building or the lot.

1 De nitions We state for reference the following de nitions. The interior of a set S in a topological space is the set of points that are contained in an open set wholly contained in S. Its interior is the set of all points that satisfy x2 y2 z2 1 while its closure is x2 y2 z2.

The exterior of a set S is the complement of the closure of S. In this sense interior and closure are dual notions. The interior of Yis the set IntY S UUYand Uis open in X.

The boundary of Yis the set BdY YXrY. A point that is in the interior of S is an interior point of S. The interior of A is 2 4 since the elements a of that interval are the only elements of A for which there is a ε 0 such that a ε a ε A.

Consider a sphere x2 y2 z2 1. If a point is neither an interior point nor a boundary point of S it is an exterior point of S. Note D and S are both closed.