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

A:

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

A:

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: 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: 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…

A:

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…