1)Let S be any setof real numbers.Prove that S° isopen. Prove that S is open if and only if S equals...

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1)Let S be any setof real numbers.Prove that S° isopen. Prove that S is open if and only if S equals its interior.2)Let S be any set of real numbers. Prove that S¯. Prove that ¯is a closed set. Prove that ¯\S° is the boundary of S.3)Let K be a compact set and let U be an open set that contains K. Prove that there is an such that if k Kthen the interval (k-,k+) is contained in U.19)Give an example of open sets U1U2...such that njUjisclosed and nonempty.27)Exhibit a countable collection of open sets Ujsuch that each opencan be written as a union of someof thesets Uj.31)Let S be a compact set and T a closed set of real numbers. Assume thatSnT=. Provethat there is a number?0 such that ||for every sSand every tT. Prove that the assertion is false if we only assume that S is closed.32)Prove that the assertion of exercise 31 is false if we assume that S and T are both open.33)LetS be any set and define, for x , dis(x,S) = inf{||}. Prove that if x ¯then disc(x,s)If x, y then prove that||||.35)Prove that every nonisolated boundary point ofa set S is an accumulationpoint of the set S.

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1) Let S be any set of real numbers. Prove that S° is open. Prove that S is open if and only if S equals its interior. ¯ ¯ ¯ 2) Let S be any set of real numbers. Prove that S . Prove that is a closed set. Prove that \S° is the boundary of S. 3) Let K be a compact set and let U be an open set that contains K. Prove that there is an such that if k K then the interval (k- ,k+ ) is contained in U. 19) Give an example of open sets U U … such that n U is closed and nonempty. 1 2 j j 27) Exhibit a countable collection of open sets U such that each open can be written as a j union of some of the sets U . j 31) Let S be a compact set and T a closed set of real numbers. Assume that SnT= . Prove that there | | is a number ?0 such that for every s S and every t T. Prove that the assertion is false if we only assume that S is closed. 32) Prove that the assertion of exercise 31 is false if we assume that S and T are both open. ¯ | | 33) Let S be any set and define, for x , dis(x,S) = inf{ }. Prove that if x then | | | | disc(x,s) If x, y then prove that . 35) Prove that every nonisolated boundary point of a set S is an accumulation point of the set S.

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David · Accepted Answer

1
Solution 1. Let x ∈ S◦. Then some neighborhodd B(x, �) is contained in S. We want to show prove that N(x, �) is
contained in S◦. Let y ∈ B(x, �). Then since a neighborhood is an open set, we have a δ > 0 such that
B(y, δ) j B(x, �). And since B(x, �) ⊆ S, we conclude that B(y, δ) ⊆ S. So y ∈ S◦. Since this is true for arbitrary
y ∈ B(x, �) we have B(x, �) ⊆ S◦.
Next, assume that S is open. Let x ∈ S. Then we have that for r > 0 we have B(x, r) ⊂ S. This shows that x ∈ S◦.
This shows that S ⊂ S◦. Also by definition of S◦ we have S◦ ⊂ S, and hence S = S◦.
Now assume S = S◦. We have to prove that S is open. Note that since S = S0 for every point x ∈ S we have
B(y, r) ⊆ S, since this point x is also in the interior. Hence S is open.
Solution. 2 If x ∈ S then clearly for any δ > 0, (x− δ, x+ δ) ∩ S contains atleast {x} and is therefore not empty.
So, x ∈ S̄.
We prove that S̄ is closed by proving R \ S̄ is always open. Now if x ∈ R \ S̄, then ∃ δ > 0 : (x− δ,

1)Let S be any setof real numbers.Prove that S° isopen. Prove that S is open if and only if S equals its interior.2)Let S be any set of real numbers. Prove that S¯. Prove that ¯is a closed set. Prove...

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