ONT Re: Topology
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| 1. Topological Spaces
|
| 1.5. Interior and Boundary (cont.)
|
| There is one other operator which occurs frequently enough to justify
| its definition. The 'boundary' of a subset A of a topological space
| X is the set of all points x which are interior to neither A nor X~A.
| Equivalently, x is a point of the boundary iff each neighborhood of
| x intersects both A and X~A. It is clear that the boundary of A is
| identical with the boundary of X~A. If X is indiscrete and A is
| neither X nor void, then the boundary of A is X, while if X is
| discrete the boundary of every subset is void. The boundary
| of an interval of real numbers, in the usual topology for
| the reals, is the set whose only members are the endpoints
| of the interval, regardless of whether the interval is open,
| closed, or half-open. The boundary of the set of rationals,
| or the set of irrationals, is the set of all real numbers.
|
| It is not difficult to discover the relations between
| boundary, closure, and interior. The following theorem,
| whose proof we omit, summarizes the facts.
|
| 10. Theorem. Let A be a subset of a
| topological space X, and let b(A)
| be the boundary of A. Then:
|
| 1. b(A) = A˜ |^| (X~A)˜ = A˜ ~ A°.
|
| 2. X ~ b(A) = A° |_| (X~A)°.
|
| 3. A˜ = A |_| b(A).
|
| 4. A° = A ~ b(A).
|
| A set is closed if and only if it contains its boundary and
| is open if and only if it is disjoint from its boundary.
|
| JLK, Gen Top, pages 45-46.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤