ONT Re: Topology
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| 1. Topological Spaces
|
| 1.7. Relativization, Separation (cont.)
|
| Three theorems on separation will be needed in the sequel.
|
| 17. Theorem. If Y and Z are subsets
| of a topological space X and both
| Y and Z are closed or both are open,
| then Y ~ Z is separated from Z ~ Y.
|
| Proof. Suppose that Y and Z are closed subsets of X. Then Y and Z
| are closed in Y |_| Z and therefore Y~Z = ((Y |_| Z) ~ Z) and Z~Y
| are open in Y |_| Z. It follows that both Y~Z and Z~Y are open
| in (Y~Z) |_| (Z~Y), and since they are complements relative to
| this set both are closed in (Y~Z) |_| (Z~Y). Consequently
| Y~Z and Z~Y are separated. A dual argument applies to
| the case where both Y and Z are open in X. þ
|
| 18. Theorem. Let X be a topological space
| which is the union of subsets Y and Z
| such that Y ~ Z and Z ~ Y are separated.
| Then the closure of a subset A of X is the
| union of the closure in Y of A |^| Y and the
| closure in Z of A |^| Z.
|
| Proof. The closure of a union of two sets
| is the union of the closures, and hence:
|
| A˜ = (A |^| Y)˜ |_| (A |^| Z~Y)˜.
|
| Consequently:
|
| A˜ |^| Y = ((A |^| Y)˜ |^| Y) |_| ((A |^| Z~Y)˜ |^| Y).
|
| The set (Z~Y)˜ is disjoint from Y~Z,
| hence (Z~Y)˜ c Z, and it follows that:
|
| (A |^| Z~Y)˜ is a subset of (A |^| Z)˜ |^| Z.
|
| Similarly:
|
| A˜ |^| Z is the union of (A |^| Z)˜ |^| Z
|
| and a subset of (A |^| Y)˜ |^| Y.
|
| Consequently:
|
| A˜ = (A˜ |^| Y) |_| (A˜ |^| Z)
|
| = ((A |^| Y)˜ |^| Y) |_| ((A |^| Z)˜ |^| Z)
|
| and the theorem is proved. þ
|
| 19. Corollary. Let X be a topological space
| which is the union of subsets Y and Z
| such that Y~Z and Z~Y are separated.
| Then a subset A of X is closed (open)
| if A |^| Y is closed (open) in Y
| and A |^| Z is closed (open) in Z.
|
| Proof. If A |^| Y and A |^| Z are closed in Y and Z respectively,
| then, by the preceding theorem, A is necessarily identical with its
| closure and is therefore closed. If A |^| Y and A |^| Z are open
| in Y and Z resepctively, then Y |^| X~A and Z |^| X~A are closed
| in Y and in Z, and hence X~A is closed and A is open. þ
|
| JLK, Gen Top, pages 52-53.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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