Prove by any method that the interval (-1, 2] is not a compact subset of R. Explain each step in your argument.

Q: 1

A: Consider the provided question, We have to prove by using only definition of a compact set, that (0,…

Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: Find an open cover of the interval (−∞, 3]. Justify why that open cover has no finite subcover.

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Q: Show that |qbu)|&x + 7元? the closed on interval [-元,T」 fo1 -π2 2くだ gluen that glven gux) for

A: Given g (x) = x +x² To find the maximum value of g(x). We can differentiate the function and equate…

Q: Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property

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Q: Show without using the Heine-Borel theorem that the line R is not a compact set in R ^ 2

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Q: Every Hausdorff space is hereditarily Hausdorff.

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Q: Find the limit point of the open interval (a, b) in IR, where a

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Q: ) Find the fomily of the equ ation of all lines in 3-space perpen diuler to 2x -y +2u u Given line

A: Family of lines contains parameters.

Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

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Q: 1. Use the Intermediate Value Theorem to prove that the equation 3x - 4x? = 3 is solvable on the…

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Q: A completely regular space is also T1-space 1 True 2 False

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Q: PROVE THE FOLOWING: PROVE thtAT IF AND ONLY %3D IF is REAL

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Q: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

A: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

Q: 4).ェ Pare subset, -f y such that CeD than fic) is not eD, then the Convers %3D そイue.

A: Solution As it's mention C=D and f^-1 same , so each and every element from  f inverse we get equal.…

Q: Show that R is not compact.

A:

Q: 13) Use the Intermediate Value Theorem to show that the polynomi: zero in the given interval. g(x) =…

A:

Q: Show that the real line is a metric space.

A: Answer is mentioned below

Q: Use the Heine-Borel theorem to give an example of a set not compact in R with the usual topology…

A: An example of a set not compact in R with the usual topology and a non-compact set R2 with the usual…

Q: Suppose H is an inner |(x,y)| ≤ 11x11-1111 product space. Show Ужуєн.

A: Given: H is an inner product space. To prove: x,y≤x·y, ∀x,y∈H

Q: State true or false with brief justification: L1[a,b] is a reflexive space

A: We use the Theorem: Let X be a separable normed space. If the dual space X* of X is non-separable,…

Q: Show that in R with the usual topology, the only open sets connected by trajectories are the open…

A: Given that R with the usual topology  A collection of of subsets of R which can be expressed as a…

Q: True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all real numbers with the…

A: The given statement is  Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with…

Q: Hew.b prove that CAXB)-Ais atways-Zero

A: Topic:- vector  algebra

Q: Let (X, CA) be a measurable space, f: X (-00, ws) and 1: X (-∞, ∞0) are measurable. Prove that the…

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Q: ☺ show dhat 2) G(x,x) => Uola) Uorlo) were G is green funcfion

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Q: Show that f: R → R defined as f(a) = 3a³- 4 is a one to one %3D iuicuon.

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Q: 2|A:k||2|| Bx:||2) < || |||B||F k%3D1

A: Please check the answer in next step

Q: Use the definition to prove that {(-

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Q: show that d connot be obtained from a norm.

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Q: Show that K, is not planar.

A: To prove: K5 is not a planar graph. Proof:  This is proved by the method of contradiction. We use…

Q: 2. Explain why there does not exist a measure Space (X,S, M) with the pro porty %3D

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Q: Q7: Prove that (R", ||-|| ) is complete metric space.

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Q: Iwant to define these Poin (Defined) 1-) sobolve spaces 2-) Reflexive real Banach SPace

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Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

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Q: Prove that 8. V57 lies on the unit circle. 8.

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Q: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: topological spaces X and Y,

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Q: 4. Use the least upper bound property of R to prove that every nonempty subset of R that is bounded…

A: Solution

Q: Find all triples of integers (x, y, z) that satisfy x² + 2y² = z2.

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Q: Give all boundary points of D.  Is D open, closed or bounded? Justify all your answers.

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Q: Is any discrete space complete? Give an example to substantiate your answer.

A: A metric space is complete if and only if every Cauchy sequence is convergent.

Q: What theorem shows that GTY = GUX? Osss O SAS O AAS O ASA

A: We have to check

Q: Exhibit an open cover of the interval (1, 2] that has no finite subcover.

A: An open cover of A is a collection of open subsets of A whose union is all of A and that a finite…

Q: 28

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Q: Ihow do you bnd the imit of as x approaches 0?

A: The limit of x+1-1x as x approaches to 0

Q: Show that X is Hausdorff if and only the diagonal A = {(x,x) | x € X} is closed in X x X.

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Q: Show that the real number R with d(x,y)=|x-2y| are not a metric space

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Q: Find a bijection between the interval (−1, 1) and R.

A: