ONT Re: Topology
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| 1. Topological Spaces
|
| 1.4. Closure (cont.)
|
| A 'closure operator' on X is an operator which assigns to each
| subset A of X a subset A^c of X such that the following four
| statements, the 'Kuratowski closure axioms', are true.
|
| a. If 0 is the void set, 0^c = 0.
|
| b. For each A, A c A^c.
|
| c. For each A, A^cc = A^c.
|
| d. For each A and B, (A |_| B)^c = A^c |_| B^c.
|
| The following theorem of Kuratowski shows that these four statements
| are actually characteristic of closure. The topology defined below
| is the topology 'associated' with a closure operator.
|
| 8. Theorem. Let ^c be a closure operator on X, let !F! be the family
| of all subsets A of X for which A^c = A, and let !T! be the family
| of complements of members of !F!. Then !T! is a topology for X,
| and A^c is the !T!-closure of A for each subset A of X.
|
| Proof. Axiom (a) shows that the void set belongs to !F!, and (d) shows that
| the union of two members of !F! is a member of !F!. Consequently the union
| of any finite subfamily (void or not) of !F! is a member of !F!. Because of
| (b), X c X^c, so that X = X^c, and the union of the members of !F! is then X.
| In view of theorem 1.4, it will follow that !T! is a topology for X if it is
| shown that the intersection of the members of any non-void subfamily of !F! is
| a member of !F!. To this end, first observe that, if B c A, then B^c c A^c,
| because A^c = ((A ~ B) |_| B)^c = (A ~ B)^c |_| B^c. Now suppose that !A!
| is a non-void subfamily of !F! and that B = |^|{A : A in !A!}. The set B is
| contained in each member of !A!, and therefore B^c c |^|{A^c : A in !A!} =
| |^|{A : A in !A!} = B. Since B c B^c, it follows that B = B^c and B in !F!.
| This shows that !T! is a topology, and it remains to show that A^c is A˜,
| the !T!-closure of A. By definition, A˜ is the intersection of all the
| !T!-closed sets, that is, the members of !F!, which contain A. By axiom
| (c), A^c in !F!, and hence A˜ c A^c. Since A˜ in !F! and A˜ contains A
| it follows that A˜ contains A^c and hence A˜ = A^c. þ
|
| JLK, Gen Top, page 43.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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