pdf. Every nonempty proper subset of Ris either open or closed.

Q: Let S be a nonempty subset in R". Prove or disprove by a counterexample that Span S = (S+)+

A: We have to prove or disprove by counterexample that Span S=S⊥⊥

Q: The set G = :a is closed under addition %3D 0 2a

A: Use the closure property of addition.

Q: be a closed subset of space Xa

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Q: Let X = {a, b, c, d, e, f, g} and B = {{a,c, e, f}, {b, c, d, f,g}, {c}, {f}}. Show that B is a base…

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Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

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Q: Let X be any set with the discrete topology. Show that X is normal.

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Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Prove or disprove. A closed and bounded subset of e' is compact.

A: PROVE OR DISPROVE :-  A closed and bounded subset of l1 is compact.

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: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection ?…

A: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection?…

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly?…

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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: Let A be a subset of a topological space X. Show that bd(A) = Ø iff A is open and closed in X.

A: We need to prove the statement.

Q: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

Q: (a) Investigate whether d: RxRR delined as d(z, u) = (- u) is a metrie on the set ol all real…

A: We investigate given distance function is metric on R or not

Q: Every bounded domain subset of D containing

A: Set theory

Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

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: Prove that T is the discrete topology for X iff every point in X is an open set

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Q: be an open subset in C and f

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Q: f (UMEGM) = UME¢ f (M)where C is a nonempty collection of subsets of C.

A: Set theory

Q: Every nonempty class C has an E-minimal element.

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

A: Solution

Q: Is p continuous? Is p an open map?

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Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

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Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

Q: Let A be a nonempty subset of R that is bounded below. Define the set B = {b: b is a lower bound of…

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Q: Let {A :A E A} be a family of subsets of IR. Show that • When A is finite, we have

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Q: For each matrice A, show for definition norm A we can write: |A| + -1

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Q: If A and B are nonempty bounded subsets of R with A C B, prove that sup A < sup B.

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Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

Q: Let A be a subset of some universe U. Prove that AN A° = Ø.

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Q: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

Q: a. Show that the null-space N(A) is a convex set. Is it true for every A?

A: As both questions are of different type,we shall solve first Que only. For other questions kindly…

Q: Give an example of a closed subset S of R2 such that conv S is not closed.

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Q: df. Every nonempty proper subset of R is either open or closed.

A: False

Q: Suppose f(z) is holomorphic in an open connected set N. Can g(z) = f(z) be holomorphic in N? Explain…

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Q: 3) let X be discrete topology and Ac X, find A ?

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Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

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Q: Every proper simply connected oper equivalent to an open subset of the

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Q: Show that X is connected if and only if the only clopen subsets are Ø and X.

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Q: Let (X7) be a topological Space and Show that A is open and closed m (xx) iff %3D

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Q: Exercise 60 Let p and q be two points in a topological space X. Show that there exists an open set U…

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Q: Show That R is Banach Sface.

A: To Show -

Q: A relationR is called a partially ordered set if R is: .a reflexive .b antisymmetric .C transitive…

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Q: Show that the subset (a, b) of R is homeomorphic with R.

A: Recall that every nonempty open interval (a,b) is homeomorphic to the nonempty interval (c,d).

Q: u (B) = inf{lal: a is a covering of B by closed boxes} for all BCR" صواب İhi

A: True

Q: Show that there must be two distinct subsets A, BC S such that

A: Every value in S is between 0 and 100 , So the sum of any subset will be at least 0 and  at most…

Q: Let X be ametric space. If A and B are each dense open subsets of X then AOB is dense open subset of…

A: First we have shown the intersection is open. Then we use the property that if a set intersect every…

Q: Let A be a subset of R and let c€ R. Explain what it means to say that c is a cluster point of A.

A: