1. For each of the following sets, decide whether they are closed, bounded, compact (both) or neithe: (a) Q². (b) {1,2,3} x (0, 1].

Q: 16. A topological space (X, t) is Lindelöf if every open cover of X has a finite or countably…

A:

Q: Give an example of a set w-limit that is a minimal set.

A: To give an example of a set w-limit that is a minimal set.  Given a collection of sets , a member…

Q: 13. The interior of a set coincides with the interior of its closure..

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that a set E ⊂ ℝ is measurable iff l(I) = m∗(I ∩ E) + m∗(I ∩ E′) for any open interval I = (a,…

A: A set E ⊂ ℝ is measurable if and only if l(I) = m∗(I ∩ E) + m∗(I ∩ E′) for any open interval I =…

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

A: Matric space

Q: Q1\ Define the compact set. Then, prove that: if A is a compact subset of F , there exists a best…

A: Compact set : A set is compact if and only if every open cover has a finite subcover. There is also…

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: Let Gbe an open set in R". Two points x, y e G are said to be equivalent if they can be joined by a…

A: Since every open subset of Rn can be written of countable union of disjoint open balls and each ball…

Q: prove the Bolzano-Weierstrass theorem by using the nested interval property

A: We have to prove that the Bolzano-Weierstrass theorem by using the nested interval…

Q: If X is inner product space over real numbers then the inner product always greater than or equal…

A: The objective is to determine whether the given statement is true or false. Given statement is, " If…

Q: The definition of x being a limit point of a set A is "x € R such that V€ > 0, N₁(x)^A\ {x} ‡ Ø”.…

A: We will write the statement that  A does not have any limit points by taking the negation of the…

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: (BOunded st) show that a subset M in a nor med space Xis bounded if and only If there is a positive…

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Q: Explain the bolzano –Weierstrass theorem. Every infinite and bounded set has atleast one limit point

A: Bolzano-Weierstrass theorem:-  Every infinite and bounded set has atleast one limit point

Q: 3) #36) Hint: {[a,b): a <b} then the Borel o-algebra this complete the pr

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Q: let Aand B be b ounded setsof real numbersprove that supCA+B) =SuPAtspanB

A:

Q: EXERCISE 2: For each a/ and b/ answer these questions: (i) Sketch the set of points (directly, no…

A: (i) The parts shaded in yellow gives the required region defined by (a) (ii) bounded (iii) closed…

Q: For which spaces ++ is it true that covery single ton set is closed

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Q: prove that i } X is a Coun table set and P: X>Y is asurjecbive then Y is Countable

A: Note: X denotes cardinality of X, that is number of elements in the set X. If X is countable means…

Q: What is the norm of a partition of a closed interval?

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Show the the Sar induce

A: According to the given information, it is required to show that equivalent norm on a vector space X…

Q: Exercise 2.7. Give an example of a topological space and a collection of open sets in that…

A:

Q: Suppose that you have a covering of a sphere by finitely many( closed) hemispheres. Prove that there…

A: Given that the sphere is covering by finitely many (closed) hemispheres.

Q: Define and discuss the density property then prove that set of rational numbers and set of…

A:

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: Theorem. Show that Closed subsets of compact sets are compact.  Need handwriting solution only.

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Q: Determine in set notation where Continuas nuuas S! x^~X + E =X)

A: Given , f(x) = 3 + x2 - √x We have to determine where the function is continuous

Q: (2) If SC R" prove that the collection of isolated points of S is count- able (hint: use a covering…

A: To show  that  collection of isolated points of a  subset    of    is countable. We  can  use Open…

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: (c) Find an open cover of D that does not have a finite sub-cover. Prove it. (d) Suppose UE is an…

A: A subset is compact if every open cover has a finite subcover.

Q: Are the following true or false? If true, prove. If false, give a counterexample. (a) The…

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Q: Question No 1 (8). Prove that the discrete topology on an uncountable set does not Satisty the…

A: Note- Let X be a topological space, then X satisfies the second axiom of countability or is second…

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: True False An open interval is not homeomorphic to either a closed interval or a half open interval.…

A:

Q: Bölom 2 1) Let is be defined E= {p€ Q:2 <p^< 3}. as Show that the se t E is closed and bounded in Q.…

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Q: Show that topologically completeness is weak hereditary property.

A: In this question, we have to show that completeness is a weak hereditary property.

Q: Which characteristic/s implies(y) that a set S of real numbers is closed? S has no accumulation…

A: To choose correct option from given question.

Q: A Moving to the next question prevents changes to this answer. Question 2 Let AcR be bounded and set…

A: Choose the correct