i) show that a closed interval is a closed set;

Q: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

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Q: Find an example of a nonempty set satisfying: • Open, every point is a limit point • Closed, every…

A: Just because a nonempty open set contains only limit points does not mean that the set contains all…

Q: i. Provide an example to prove that the arbitrary intersection of open sets is not an open set ii.…

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Q: Give an example to show arbitrary union of open sets is again open. Is the result true for arbitrary…

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Q: Let S={(a,b) : a, b \in Q} be the collection of open intervals in R with rational endpoints. Find…

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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: If a class interval has NO frequency, it is said to be O a. Exclusive Class interval O b. Limited…

A: Frequency of class interval : Number of observations falling within a particular class interval is…

Q: What is the class width (i) if one of the class interval is 18-26?

A: If class interval given is 18-26 The class width is difference between upper interval and lower…

Q: What is meant by Intersection of sets?

A: Let A and B be the two sets. Then the set of all common elements of A and B is called the…

Q: True or False The intersection of two sets is always a subsetof their union

A: To check the statement

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

Q: Describe the set as a union of finite or infinite intervals {x : Ix - 4 I > 2}

A: x:x-4>2

Q: What is the intersection of sets A and B?

A: The intersection of sets is a set which include the common of the two sets A and B. It is…

Q: The continuous image of a connected set is connected. That is, if X is connected and f: X→ Y is…

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Q: E is closed. - E is complete, i.

A: It is given that X is a complete metric space and E is a subset of X. We have to prove that the…

Q: Give example open set is compact

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Q: Define two sets, call them A and B, that makes the following statement true; if it is impossible to…

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Q: A line is a collection of _______ along a _______ path that extends endlessly in bothdirections.

A: A line is a collection of _______ along a _______ path that extends endlessly in both directions.

Q: What is meant by the term open interval in sets? Give example.

A: Open interval means it does not include the end values of the given set.

Q: Prove the inequality theorem 6 which states that the sum of the lengths of any two sides of a…

A: Consider a triangle ABC as shown below.

Q: Describe the set S = x :12 x − 3> 4in terms of intervals

A: Given: S=x:12x-3>4 To Describe: Given set in the intervals.

Q: Show that any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

Q: Show that the set given in the image below is non convex. Please prove the result mathematically…

A: A set is said to be convex if for any two points in the set the line segment joining those points is…

Q: Describe the set as a union of finite or infinite intervals. {x : |x − 4| > 2}

A: can be written as x-4>2 or x-4<-2

Q: By considering the collection of intervals {(-) : n € N} show that the intersection of an infinite…

A: Let  Sn=-1n,1n :n∈N      where N is set of natural number. for n=1,2,3,...... S1=-1,1…

Q: Prove that the unoin of a finite number of bounded sets is bounded.

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Q: Show that the Cantor Set contains no open intervals

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Q: Tet X and Y be infinite sets prove that XUY is infinite

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Q: Prove that interval [0,1] is uncountable.

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Q: Prove that derived set of a bounded set is also bounded.

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Q: Prove that the boundary points of an open interval (a, b) CR are the points a and b.

A: A point a∈ℝ is said to be boundary point of a set A⊂ℝ, if Ba,r∩A≠∅ and Ba,r∩Ac≠∅ for every…

Q: show whether X *y= union, unit element Pand elenent properties exist on the set

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Q: Show that any subset of bounded set is bounded

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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: How to Apply the Archimedes'principle to prove that a set is empty

A: Consider the given information. the Archimedes' principle is defined as, Property states that for…

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: The union of a infinite number of closed set need not be a Closed set Open Set Both open and close…

A: The union of infinite number of closed sets is

Q: Describe the set as a union of finite or infinite intervals.  {x : |x 2 -1 | > 2}

A: can be written as x^2-1>2 or x^2-1<-2

Q: Explain the idea of cardinality. When is a set denumerable? Show that the union of two denumerable…

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Q: جليلة Let I = [= 1,₂1] ISR interval w that I is negligible Set.

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Q: Using the definition of an open set, prove that the interval (0, 1) is an open set.

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Q: 2. Show that the set N has no limit point.

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Q: show vacuous proof of statement that "empty set is a subset of any set"

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Q: Show wheatear an open interval (m, n) is a union of a countable collection of closed sets or not.

A:

Q: Any two closed intervals has the different cardinality

A: we know that any two closed intervals has the same cardinality

Q: In any metric space, the only open and closed set at any time is the empty set and X only True False

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Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

Q: 28

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Q: A finite set .A is open .B has no limit point .C has a limit point .D uncountable

A: Option (B) is correct.

Q: Show that the Nested Interval Property together with the Archimedean Principle implies the Axiom of…

A: