Which sets are compact? The union of [0, 1] and [2, 3]

Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

Q: 6. Show that the span of this set is all of R?. {}}

A: Let S=v1,v2,...,vn be a set of vectors in a vector space V(F) . The vector space V is spanned by S,…

Q: True or false? For all sets A and B,  (A∪B)'=A'∪B' This can be determined by constructing a…

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Q: If a < b, then [a,b] CR does not have con- tent 0. In fact, if U1,. . .,Un) is a finite cover of…

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Q: Show that an infinite intersection of closed sets F, k = 1, 2, 3, . .., in a metric space (S, d) is…

A: first we will proof that an infinite union of Open sets in metric space ( S, d) is open using the…

Q: Find the dimension of the set S= {(2, 4), (4, 8)}?

A: The dimension of the set S can be find out by finding the linearly independent vectors in it.…

Q: The boundary of a set is equal to the boundary of the complement True O False O

A: A point of a set is called boundary points if any neighbourhood of this point contains point of the…

Q: If C1, C2 are connected subsets of R, then the product C, xC, is a connected subset of R?,

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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: 3) Consider the subsets [0, 1] and [3,4] of R, the set of all real numbers with the euclidean…

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Q: Show that a set {V1, ..., Vp} in R" is affinely dependent when p > n + 2.

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Q: Can a set and it's Complement be both nowhere dense in R? A Ac=R-A

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Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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Q: Sketch the following set and show if it is open, closed, and bounded or unbounded. 0 < arg(2) < 7/4…

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Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: Which of the following sets spans R? None of these O {(-3,0,2)} O ((-3,0,2),(0,0,0)} O…

A: As per our guidelines only one question can be solved. Solution for the 1st question is given below…

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

A: Solution

Q: Describe the set {TER²: doo (r,(0,0)) = 1}.

A: Given: Set x∈R2:d∞(x,(0,0))=1 We have to describe the set.

Q: determine whether set s={(2,1),(-1,2)} spans R^2

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Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

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Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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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: Show that R³ is spanned by S = {(1,1,1), (1, –1, 0), (0, 1, – 1)}.

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Q: union of a collectionm of connected sets that have a point in common is conne

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Q: Let (X, M, u) be a measure space wE

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

Q: given: set A = {a,b,c,d} into set B = {1,2,3,4}. which set of ordered pairs represent a bijection?

A: Consider the set S={(a,1), (b,2), (c,3), (d,4)} Here each element of A has exactly one image in B…

Q: 2. Show that the set 3 S = 3 2 2 2 2 spans all of R³ or find an element of R' not belonging to…

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Q: Show that R³ is spanned by S = {(1,1,1), (1, – 1, 0), (0, 1, – 1)}.

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Q: Define the Cartesian product of sets X and Y? How is this Cartesian product denoted?

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Q: List all the possible subsets of set G, where G = {-7,m,z}- %3D

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Q: OShow that the set (1,2] is not compact 2Show that the set [1,2) is not compact 3Show that the set…

A: 1.To show that (1,2] is not compact

Q: Describe the set {TER²: doo (r,(0, 0)) = 1}.

A: To Describe: the set {x∈ℝ2: d∞(x,(0,0))=1}.

Q: A set L cR is convex if:

A: In this question, we have to find the correct option for the convex set L⊂Rn  here we use the direct…

Q: If X = {u,v} what is the power set of X? (in answering this question, use {} as the null set, and do…

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Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: Show that the interval [0,1] is a closed set in R

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Q: Which sets are compact? The set of integers L

A: The objective is to determine whether the set of integers ℤ is compact or not.

Q: The set {0,3} with the usual metric is connected as union of connected sets {0} and {3} True False

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Q: If fis a bijection, find the sets X and Y.

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Q: • Show that if A is compact and Bis closed, and d(A, B) = 0, then ANB÷Ø.

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Q: EX LeT X = {!,2,3,415,6} T- Ex,¢,てり,E4,しろ4), [ス456 Topology.?

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Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: Determine if the set S is span set of %3D R2x2. Solution:

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

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Q: Let the non-empty set S = {x - yv3:x,y ER}, show that S is closed'under multiplication then find the…

A: We have to show that the given set S=x-y3:x,y∈R is closed under multiplication and also find the…

Q: Show that R3 is spanned by S = {(1,1,1), (1, –1,0), (0, 1, – 1)}

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Q: Which of the following sets spans R°? None of these {(-3,0,2)} O {(-3,0,2),(5,1,0),(0,1,1)}…

A: The solution is given as

Q: Is it true that (A×A)∖(B×B)=(A∖B)×(A∖B) for any two sets A, B? If it is true, prove it. Otherwise,…

A: Explanation of the answer is as follows

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.