Suppose A and B are closed subsets of R. Show that A ∩ B and A...

Suppose A and B are closed subsets of R. Show that A ∩ B and A ∪ B are closed.

Expert Solution

Colby Messinger answered 2 years ago

Suppose A and B are subsets of R, and deﬁne: d(A,B) = inf{|a−b| : a ∈...

Suppose A and B are subsets of R, and deﬁne: d(A,B) = inf{|a−b| : a ∈ A,b ∈ B}. (a) Show that if A∩B 6= ∅, then d(A,B) = 0. (b) If A is compact, B is closed, and A∩B = ∅, show d(A,B) > 0. (c) Find 2 closed, disjoint subsets of R (say A and B) with d(A,B) = 0

Show if A, B, C are connected subsets of X and A∩B not equal ∅ and...

Show if A, B, C are connected subsets of X and A∩B not equal ∅ and B ∩C not equal ∅, then A∪B ∪C is connected.

Let A and B be two non empty bounded subsets of R: 1) Let A +B...

Let A and B be two non empty bounded subsets of R: 1) Let A +B = { x+y/ x ∈ A and y ∈ B} show that sup(A+B)= sup A + sup B 2) For c ≥ 0, let cA= { cx /x ∈ A} show that sup cA = c sup A hint:( show c supA is a U.B for cA and show if l < csupA then l is not U.B)

6.3.8. Problem. Let f : A → B be a continuous bijection between subsets of R....

6.3.8. Problem. Let f : A → B be a continuous bijection between subsets of R. (a) Show by example that f need not be a homeomorphism. (b) Show that if A is compact, then f must be a homeomorphism. 6.3.9. Problem. Find in Q a set which is both relatively closed and bounded but which is not compact.

The goal is to show that a nonempty subset C⊆R is closed iff there is a...

The goal is to show that a nonempty subset C⊆R is closed iff there is a continuous function g:R→R such that C=g−1(0). 1) Show the IF part. (Hint: explain why the inverse image of a closed set is closed.) 2) Show the ONLY IF part. (Hint: you may cite parts of Exercise 4.3.12 if needed.)

Let f : [0, 1] → R and suppose that, for all finite subsets of [0,...

Let f : [0, 1] → R and suppose that, for all finite subsets of [0, 1], 0 ≤ x1 < x2 < · · · < xn ≤ 1, we have |f(x1) + f(x2) + · · · + f(xn)| ≤ 1. Let S := {x ∈ [0, 1] : f(x) ̸= 0}. Show that S is countable

Let A be a bounded linear operator on Hilbert space. Show that if R(A) is closed,...

Let A be a bounded linear operator on Hilbert space. Show that if R(A) is closed, so is R(A*)

Given the following subsets of R: A = R\Q = {x ∈ R|x not in Q}...

Given the following subsets of R: A = R\Q = {x ∈ R|x not in Q} B = {1, 2, 3, 4} C = (0, 1] D = (0, 1] ∪ [2, 3) ∪ (4, 5] ∪ [6, 7] ∪ {8} (a) Find the set of limit points for each subset when considered as subsets of RU (usual topology on R). (b) Find the set of limit points for each subset when considered as subsets of RRR (right-ray topology on...

1. Let a < b. (a) Show that R[a, b] is uncountable

Consider the following collections of subsets of R: B1 ={(a,∞):a∈R}, B2 ={(−∞,a):a∈R}, B3 ={[a,∞):a∈R}, B4 =...

Consider the following collections of subsets of R: B1 ={(a,∞):a∈R}, B2 ={(−∞,a):a∈R}, B3 ={[a,∞):a∈R}, B4 = {[a, b] : a, b ∈ R}, B5 = {[a, b] : a, b ∈ Q}, B6 ={[a,b]:a∈R,b∈Q}. (i) Show that each of these is a basis for a topology on R. (ii) What can you say about the corresponding topologies T1,...,T6, eg, are any of the topologies the same, are any comparable, are any equal to familiar topologies on R, etc?

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