ONT Re: Topology
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| 1. Topological Spaces
|
| 1.4. Closure
|
| The 'closure' (!T!-closure) of a subset A of a topological space
| (X, !T!) is the intersection of the members of the family of all
| closed sets containing A. The closure of A is denoted by A˜, or
| by Ã. The set A˜ is always closed because it is the intersection
| of closed sets, and evidently A˜ is contained in each closed set
| which contains A. Consequently A˜ is the smallest closed set
| containing A and it follows that A is closed if and only if
| A = A˜. The next theorem describes the closure of a set
| in terms of its accumulation points.
|
| 7. Theorem. The closure of any set
| is the union of the set and the
| set of its accumulation points.
|
| Proof. Every accumulation point of a set A is an
| accumulation point of each set containing A, and is
| therefore a member of each closed set containing A.
| Hence A˜ contains A and all accumulation points of A.
| On the other hand, according to the preceding theorem,
| the set consisting of A and its accumulation points is
| closed and it therefore contains A˜. þ
|
| The function which assigns to each subset A of a topological
| space the value A˜ might be called the closure function, or
| closure operator, relative to the topology. This operator
| determines the topology completely, for a set A is closed
| iff A = A˜. In other words, the closed sets are simply
| the sets which are fixed under the closure operator.
|
| It is instructive to enquire: Under what circumstances is an operator
| which is defined for all subsets of a fixed set X the closure operator
| relative to some topology for X? It turns out that four very simple
| properties serve to describe closure. First, because the void set
| is closed, the closure of the void set is void; and, second, each
| set is contained in its closure. Next, because the closure of each
| set is closed, the closure of the closure of a set is identical with
| the closure of the set (in the usual algebraic terminology, the closure
| operator is idempotent). Finally, the closure of the union of two sets is
| the union of the closures, for (A |_| B)˜ is always a closed set containing
| A and B, and therefore contains A˜ and B˜ and hence A˜ |_| B˜. On the other
| hand, A˜ |_| B˜ is a closed set containing A |_| B and hence also (A |_| B)˜.
|
| JLK, Gen Top, pages 42-43.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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