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 R).

Expert Solution

Colby Messinger answered 8 months ago

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?

Given the function u(p,q,r)=((p-q)/(q-r)), with p=x+y+z,q=x-y+z, and r=x+y-z, find the partial derivatives au/ax=, au/ay=, au/az=

Let X be the set of all subsets of R whose complement is a finite set...

Let X be the set of all subsets of R whose complement is a finite set in R: X = {O ⊂ R | R − O is finite} ∪ {∅} a) Show that T is a topological structure no R. b) Prove that (R, X) is connected. c) Prove that (R, X) is compact.

Let x,y ∈ R satisfy x < y. Prove that there exists a q ∈ Q...

Let x,y ∈ R satisfy x < y. Prove that there exists a q ∈ Q such that x < q < y. Strategy for solving the problem Show that there exists an n ∈ N+ such that 0 < 1/n < y - x. Letting A = {k : Z | k < ny}, where Z denotes the set of all integers, show that A is a non-empty subset of R with an upper bound in R. (Hint: Use...

5. Given the function y = q(x) = (x^2)/(x-1) a. What is the domain of q(x)?...

5. Given the function y = q(x) = (x^2)/(x-1) a. What is the domain of q(x)? b. What are the vertical asymptotes? c. What are the horizontal asymptotes? d. Where is q(x) increasing/decreasing (draw a line and specify by intervals – be sure to include points where q isn’t defined)? e. Where is q(x) concave up/down ((draw a line and specify by intervals – be sure to include points where q isn’t defined)? f. Find rel max/min. g. Find inflection...

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.

Using the simplex method, maximize revenue given the following R = 40q + 20z q +...

Using the simplex method, maximize revenue given the following R = 40q + 20z q + z ≤ 50 15q + 20z ≤ 200 q,z ≥ 0

Profit = Revenue - Cost P(x) = R(x) - C(x) Given: R(x) = x^2 - 30x...

Profit = Revenue - Cost P(x) = R(x) - C(x) Given: R(x) = x^2 - 30x Given: C(x) = 5x + 100 X is hundreds of items sold / P, R, C are in hundreds of dollars (1) Determine the Initial Cost? (2) Determine the maximum Profit and number of items required for that profit? (3) Determine the maximum Revenue and number of items required for that revenue? (4) Find the break even points, P(x) = 0. What do the...

X and Y are subsets of a universal set U. Is the following statement true or...

X and Y are subsets of a universal set U. Is the following statement true or false? Support your answer with a venn diagram. X - (Y u Z) = (X - Y) n (A - C)

For the following exercises, the revenue generated by selling x items is given by R(x) = 2x2 + 10x. Find R′(10) and interpret.

For the following exercises, the revenue generated by selling x items is given by R(x) = 2x2 + 10x.Find R′(10) and interpret.

Question