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Math 161, Sheet 3: the Topology of the Continuum

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Math 161, Sheet 3: the Topology of the Continuum MATH 161, SHEET 3: THE TOPOLOGY OF THE CONTINUUM In this sheet we give the continuum C a topology. Roughly speaking, this is a way to describe how the points of C are `glued together'. Definition 3.1. A subset of the continuum is closed if it contains all of its limit points. Theorem 3.2. The sets Ø and C are closed. Theorem 3.3. A subset of C containing a finite number of points is closed. Definition 3.4. Let X be a subset of C. The closure of X is the subset X of C defined by: X = X [ fx 2 C j x is a limit point of Xg: Theorem 3.5. X ⊂ C is closed if and only if X = X. Theorem 3.6. The closure of X ⊂ C satisfies X = X. Corollary 3.7. Given any subset X ⊂ C, the closure X is closed. Definition 3.8. A subset U of the continuum is open if its complement C n U is closed. Theorem 3.9. The sets Ø and C are open. The following is a very useful criterion to determine whether a set of points is open. Theorem 3.10. Let U ⊂ C. Then U is open if and only if for all x 2 U, there exists a region R such that x 2 R ⊂ U. Corollary 3.11. Every region R is open. Every complement of a region C n R is closed. Corollary 3.12. Let a 2 C. Then the sets fx j x < ag and fx j a < xg are open. Theorem 3.13. Let U be a nonempty open set. Then U is the union of a collection of regions. Exercise 3.14. Do there exist subsets X ⊂ C that are neither open nor closed? Theorem 3.15. Let fXλg be an arbitrary collection of closed subsets of the continuum. T Then the intersection λ Xλ is closed. Theorem 3.16. Let U1;:::;Un be a finite collection of open subsets of the continuum. Then the intersection U1 \···\ Un is open. Exercise 3.17. Is it necessarily the case that the intersection of an infinite number of open sets is open? 1 Corollary 3.18. Let fUλg be an arbitrary collection of open subsets of the continuum. Then S the union λ Uλ is open. Let X1;:::;Xn be a finite collection of closed subsets of the con- tinuum. Then the union X1 [···[ Xn is closed. Theorem 3.9 and Corollary 3.18 say that the collection T of open subsets of the contin- uum is a topology on C, in the following sense: Definition 3.19. Let X be any set. A topology on X is a collection T of subsets of X that satisfy the following properties: 1. X and Ø are elements of T . 2. The union of an arbitrary collection of sets in T is also in T . 3. The intersection of a finite number of sets in T is also in T . The elements of T are called the open sets of X. The set X with the structure of the topology T is called a topological space1. Theorem 3.13 says that every nonempty open set is the union of a collection of regions. This necessary condition for open sets is also sufficient: Theorem 3.20. Let U ⊂ C be nonempty. Then U is open if and only if U is the union of a collection of regions. Definition 3.21. A topological space X is discrete if every subset of X is open. Exercise 3.22. Find a realization of the continuum that is discrete. Must every realization be discrete? Definition 3.23. Let A and B be nonempty disjoint subsets of a topological space X. We say that A and B are separated if each contains no point of the closure of the other, i.e. A \ B = Ø and A \ B = Ø. Theorem 3.24. Let ab be a region in C. Then the sets ab and ext ab are separated. Definition 3.25. Let X be a topological space. X is disconnected if it may be written as the union X = A [ B of two separated sets. X is connected if it is not disconnected. Exercise 3.26. Is the continuum connected? 1The word topology comes from the Greek word topos (τoπoζ´ ), which means \place". 2.
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