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

Expert Solution

Colby Messinger answered 8 months ago

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.

Prove: inf(A + B) = infA + infB, where A and B are nonempty subsets of...

Prove: inf(A + B) = infA + infB, where A and B are nonempty subsets of the reals and A + B = a + b for all a in A and b in B, and infA, infB represents the infimums of thee two sets with lower bounds. Please help!

in the group R* find elements a and b such that |a|=inf, |b|=inf and |ab|=2

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.

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

construct a bijective function of f:[0, inf) -> R

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...

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?

Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if...

Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if and only if ad=bc. (a) Show that R is an equivalence relation. (b) What is the equivalence class of (1,2)? List out at least five elements of the equivalence class. (c) Give an interpretation of the equivalence classes for R. [Here, an interpretation is a description of the equivalence classes that is more meaningful than a mere repetition of the definition of R. Hint:...

Question