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MATH 113, SHEET 2: THE TOPOLOGY OF THE CONTINUUM John Lind • September 27, 2013 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 2.1. A subset of the continuum is closed if it contains all of its limit points. Theorem 2.2. The sets Ø and C are closed. Theorem 2.3. A subset of C containing a finite number of points is closed. Definition 2.4. Let X be a subset of C. The closure of X is the subset X of C defined by: X = X ∪ {x ∈ C | x is a limit point of X}. Theorem 2.5. X ⊂ C is closed if and only if X = X. Theorem 2.6. The closure of X ⊂ C satisfies X = X. Corollary 2.7. Given any subset X ⊂ C, the closure X is closed. Definition 2.8. A subset U of the continuum is open if its complement C \ U is closed. Theorem 2.9. The sets Ø and C are open. The following is a very useful criterion to determine whether a set of points is open. Theorem 2.10. Let U ⊂ C. Then U is open if and only if for all x ∈ U , there exists a region R such that x ∈ R ⊂ U . Corollary 2.11. Every region R is open. Every complement of a region C \ R is closed. Corollary 2.12. Let a ∈ C. Then the sets {x | x < a} and {x | a < x} are open. Theorem 2.13. Let U be a nonempty open set. Then U is the union of a collection of regions. Exercise 2.14. Do there exist subsets X ⊂ C that are neither open nor closed? Theorem 2.15. Let T {Xλ } be an arbitrary collection of closed subsets of the continuum. Then the intersection λ Xλ is closed. Theorem 2.16. Let U1 , . . . , Un be a finite collection of open subsets of the continuum. Then the intersection U1 ∩ · · · ∩ Un is open. 1 Exercise 2.17. Is it necessarily the case that the intersection of an infinite number of open sets is open? CorollaryS2.18. Let {Uλ } be an arbitrary collection of open subsets of the continuum. Then the union λ Uλ is open. Let X1 , . . . , Xn be a finite collection of closed subsets of the continuum. Then the union X1 ∪ · · · ∪ Xn is closed. Theorem 2.9, Theorem 2.16 and Corollary 2.16 say that the collection T of open subsets of the continuum is a topology on C, in the following sense: Definition 2.19. 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 space 1 . Theorem 2.13 says that every nonempty open set is the union of a collection of regions. This necessary condition for open sets is also sufficient: Theorem 2.20. Let U ⊂ C be nonempty. Then U is open if and only if U is the union of a collection of regions. Definition 2.21. A topological space X is discrete if every subset of X is open. Exercise 2.22. Find a realization of the continuum that is discrete. Must every realization be discrete? Definition 2.23. 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 2.24. Let ab be a region in C. Then the sets ab and ext ab are separated. Definition 2.25. 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 2.26. Is the continuum connected? 1 The word topology comes from the Greek word topos (τ óπoζ), which means “place”. 2