ONT Re: Topology
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| 1. Topological Spaces
|
| 1.5. Interior and Boundary
|
| There is another operator defined on the family
| of all subsets of a topological space, which is
| very intimately related to the closure operator.
| A point x of a subset A of a topological space is
| an 'interior point' of A iff A is a neighborhood of x,
| and the set of all interior points of A is the 'interior'
| of A, denoted A°. (In the usual terminology, the relation
| "is an interior point of" is the inverse of the relation
| "is a neighborhood of".) It is convenient to exhibit
| the connection between this notion and the earlier
| concepts before considering examples.
|
| 9. Theorem. Let A be a subset of a topological space X.
| Then the interior A° of A is open and is the largest
| open subset of A. A set A is open if and only if
| A = A°. The set of all points of A which are not
| points of accumulation of X~A is precisely A°.
| The closure of X~A is X ~ A°.
|
| Proof. If a point x belongs to the interior of a set A, then x is
| a member of some open subset U of A. Every member of U is also a
| member of A°, and consequently A° contains a neighborhood of each
| of its points and is therefore open. If V is an open subset of A
| and y in V, then A is a neighborhood of y and so y in A°. Hence
| A° contains each open subset of A and it is therefore the largest
| open subset of A. If A is open, then A is surely identical with
| the largest open subset of A. Hence A is open iff A = A°. If x
| is a point of A which is not an accumulation point of X~A, then
| there is a neighborhood U of x which does not intersect X~A and
| is therefore a subset of A. Then A is a neighborhood of x and
| x in A°. On the other hand, A° is a neighborhood of each of
| its points and A° does not intersect X~A, so that no point
| of A° is an accumulation point of X~A. Finally, since A°
| consists of the points of A which are not accumulation
| points of X~A, the complement, X ~ A°, is precisely
| the set of all points which are either points of
| X~A or accumulation points of X~A, that is,
| the complement is the closure (X~A)^c. þ
|
| JLK, Gen Top, page 44.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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