7:35 AM ..ooo Sprint LTE K Back 3318 LPSnotes.pdf Chapter 6: Closed Point Sets a

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7:35 AM ..ooo Sprint LTE K Back 3318 LPSnotes.pdf Chapter 6: Closed Point Sets and Closure Definition. A point set M is closed meaass that if pis a limit point of M.then p belong to M. Exercise 6.1. Let A 10. B (0,10), C (0,10 D 1,2,3, 9, 10). R, are A, B, C, D closed sets? a. In S In S R. is (0.3) u (3.10) a cloeed set? d. Is a closed set in S- R? ia S Z? e. Is R a closed in S R? in S Z? Problem 6.1. Give an exam ple of a number ret that is not open and not cloeed. Problem 6.2. Give examples of a closed set MCR such that: e is a proper noeempty kubeet of M. Theorem 6.1. If g is a nonempty, finite collection of closed sets, then UaesGis cksed Theorem 6.2. If g is a Domempty collection of closed sets having a commoe Post, then naesG dosed. Theorem 6-3. If the open set M is a proper subset of S. them Me is doeed. Theorem 6.4 If the closed set M is a peoper subset of S, ther Me is open. Problem 6.3. Give a model of our axioua syytem in which there is a proper subert that is both open and Theorem 6.5. If H is a point art, then HUH' is ck ed. Definition, Ir H is a point set, then the set HUEr is called the closure of H and is written H Exercise 6.2. Let A 10. B (0,10), c 0,10 D (1.2.3,... 9, 10). What are C.DT Theorem 6.6. If is the collection of au ckeed tscontaining a point -t H. then noe G-H Question 6.1. The previous two theorems are used to describe as the "smallest closed set containing H. Make this description rigorous. Question 6.2 Sappoee M is a point set. Prove or disprove that a. M is closed if and only if M. (T/Y-17 Definition. If z

Explanation / Answer

Q the set of all rational number is not open and not closed in R the set of real numbers.

Sice the limit point of Q is not in Q, so it is not closed.

Q is not open since it is not the union of open intervals(any open interval (a,b) contains an irrational number)

So it is not open.

One more example, A semi open interval [0,1) is not open and not closed.

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