1.86 Prove or give a counterexample: Every open cover of a finite subset of R has a finite subcover. (Note: For the real line, the phrase finite subset does not mean the same thing as finite interval.)

Q: - Recall that for x, y € R, x - y = R is the absolute value of x-y. That is, if x-y 20 if x-y<0. x-y…

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Q: Let A be a non-empty set of real numbers which has a upremum and let B= {x : - x e A}. Show that B…

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Q: Suppose that A is not an empty set, and is a set of real numbers bounded below. Let -A be the set of…

A: Hello, thanks for your question but according to our policy I am answering the very first question.…

Q: 11. If a set has empty interior, then so does its closure. 12. The closure of a set coincides with…

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Q: Use the Heine-Borel Property from (5.1), which states that a set of real numbers is compact if and…

A: See solution below. Here we used definition of bounded and closed sets.

Q: (b) Show that the continuous image of compact set is compact but the convers need not to be true.…

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Q: the collection of all close subsets of R is called the topology of R

A: To check: Whether the statement is true or false: Given: The statement is "The collection of all…

Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: Suppose S is a nonvoid, bounded set of real numbers. Let - S denote the set {-x: ES}. Then inf(S)…

A: The answer provided below has been developed in a clear step by step manner

Q: Let ∅ be the empty set. Let X be the real line, the entire plane, or, in the three-dimensional case,…

A: Given:

Q: 1. For each of the following sets, decide whether they are closed, bounded, compact (both) or…

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Q: . Sketch the graph of the set defined by A x B where A is the closed interval [-2, 4] and B = {1, 3,…

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Q: Exercise 4. A subset A of a topological space X is called dense if cl A = X. The set Q of rational…

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Q: R.1. Use the Schröder-Bernstein Theorem to prove that the intervals [0, 1) and (1, ∞) on the real…

A: The trigonometric tangent function sprang to mind first, and a strategic linear transformation of…

Q: 3) Let X be a set with > 3 elements. Show that there exist bijective functions f :X → X and g : X +…

A: Let x=1,2,3 then, f=123132g=123321 `

Q: The set X is an open set in the ambient metric space . O True O False

A: Solution

Q: b) Prove that 3abc ≤ a³ + b³ + c³ for nonnegative a, b, c. (Hint: In the in- equality of part (a),…

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Q: ctors - Z. S. Madhi

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Q: Is the set {(x1, x2) ∈ R2; 2x1+x2 ≥ 1} convex? Justify your objection.

A: Consider the set      S = {(x1, x2) ∈ R2 : 2x1 + x2 ≥ 1} Let, the points (x1, x2), (x3, x4) be in S.…

Q: Find the supremum and infimum of the set {x € R|10/T – I > 0}, if they exist. If they exist, provide…

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Q: Let A = {a, b, c, d} and R = {(a, b), (b, c), (c, c), (a, a)} a. Find the transitive closure of R.…

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Q: Let R be the set of all real numbers with the euclidean topology. Define the set A = {(r, y) ER x R…

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Q: What is the interior of S? b. What is the boundary of S? c. Is S open, closed, or neither? All of…

A: a. Interior point is empty b. Boundary of set S is all point of set S c.  Set S is closed.

Q: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

Q: Consider the relation ? = {(a, b), (a, d), (b, a), (c, d)} on the set {a, b, c, d}.  a.  Find the…

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Q: (ii) Check whether or not each of the following interval is a neighborhood of 0 under usual topology…

A: (ii).  (iii).

Q: Which of the following sets is not closed on the real line with the usual metric? (i) The set of…

A: Option (iii) the set of rational numbers is not closed on the real line with the usual metric.

Q: Prove. For any x, y ∈ R and x < y, ⟨(x, y), ≤⟩ is not well-ordered, where (x, y) is an open interval…

A: To Prove: x,y,≤ is not well ordered, where x,y is an open set with the usual order "≤".

Q: Classify each of the following subsets of R as compact, connected, neither, or both. Justify your…

A: (a) Let      We  have to determine  whether is  compact or connected or both .    Let us  first…

Q: - Give the quadratic function fle)= x?-5x Compute flath) -f(a) | flath)-f(a)

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Q: 2. Classify each of the following sets as open, closed, neither, or both. Then, find the closure. a)…

A: classify the given sets are open and closed.

Q: Find an orthogonal set with the same span (and same number of elements) as [2] [8]} (18118)} "

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Q: Prove or disprove: All uncountable measurable subsets of ℝ have positive measure.

A: All uncountable measurable subsets of ℝ have positive measure.