Exercise 2.7. Give an example of a topological space and a collection of open sets in that topological space that show that the infinite intersection of open sets need not be open.

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: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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Q: Show in the discrete metric space: 1. All sets are open (especially single points);

A: Let X be a non-empty set and d be discrete metric on X

Q: In the discrete topology on the set X, every set is open and it also follows that every set is…

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Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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Q: Exercise 2.15. Construct uncountable sets A and B such that A\ B is: (A) finite (B) countably…

A: See the attachment

Q: 3.1 Show that an infinite intersection of closed sets E k = 1,2,3,., in a metric space (SA) is a…

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Q: Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.

A: A set is said to be infinite countable if its elements can be counted like natural number but you…

Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

Q: 2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite…

A: It is given in part (a) that (X,t) be any topological space, and the interval (x-A,x+A) is only open…

Q: show that any infinite discrete topological space is not compact

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Q: a) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

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: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

Q: Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) The sets in…

A: 2.46

Q: Prove that the quotient of a locally connected topological space is locally connected.

A: We have to prove that Let q:X→Y be a quotient map, X is a locally connected space. Show that xY is…

Q: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

A: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

Q: Provide a direct proof that a sequentially compact set must be both closed and bounded without…

A: WE GIVE A DIRECT PROOF. First, we prove that set A is closed then A is bounded.

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: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

Q: 1. (a) Define what a linear subset of a projective space is.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: 3.1 Show that an infinite intersection of closed sets F, k = 1,2, 3, ..., in a metric (S, d) is a…

A: Let {Fk}  be an arbitrary collection of closed sets . . 1  i.e. for k = 1, 2, 3, . . . . . . . . To…

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

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Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

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

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

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: 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: Hausdorff space.

A: Given: A set X with co-finite topology and  Hausdorff space. To Find: Under what condition a set X…

Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

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

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Q: Find a topology in which the point x = 5 is an open set.

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Q: Show that any infinite subset A of a discrete topological space X is not compact.

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Q: 3.1 Show that an infinite intersection of closed sets Fa, k = 1, 2,3,.., in a metric space (S, d) is…

A: The given is related to Metric spaces. The solution is explained below.

Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: 1. Show that the real line is a metric space.

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Q: A finite set in a metric space cannot have accumulation points.

A: We have to prove that A finite set in a metric space can not have accumulation (Limit) point.

Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: Show that all open balls are bounded subsets of a metric space.

A: Solution: Let M=X,d be a metric space. For an arbitrary point a∈X, the open ball Bεa where ε>0 is…

Q: Prove that in any metric space, an ɛ- eighborhood of a point is an open set.

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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: (a) A discrete space with more than one point is disconnected. (b) Any trivial space is connected…

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Q: Let (X,τ) is a topological space and A ⊆ X. If all subsets of A are closed in X, then set A cannot…

A: Given that X,τ is a topological space. Let's prove this theorem by contradiction. So  X,τ is a…

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: 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: Theorem 2.20. For any set A in a topological space X, the closure of A equals the inter- section of…

A: Let A  be a set in the topological space X. To prove that the closure of A  is the intersection of…

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: 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…

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.