Question 6. In a topological space (X,r) a subset ACX is called regular open if A=int(cl (A)) "A= A". Determine whether the family of all regular open sets is closed under arbitrary union or not.

Q: let x be a discrete topology Find ent (A) blit int (A) jet AC X and

A: Given that A⊂X where X is a discrete topology. To find: (i) ext (A).             (ii) b(A).…

Q: Let X be a normed space and aɛ X. Show that there exists an fe X' such that ||a|| = Sup { \f(a)| :…

A: Given that,                X is a normed space and a ∈ X . We have to show that there exists an f ∈…

Q: Question 3. Let , and r, be two topologies on X such that ,cr;. Construct a space on which a r-limit…

A: Given below the detailed solution

Q: Let A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

A:

Q: Question 6. In a topological space (x.r) a subset AC X is called regular open if A=int(d(A)) "A=A".…

A:

Q: Question 7. Let A=[0,1]U(1,2)nQ) be a subset in the usual topological space where Q is the set of…

A:

Q: Let (X, Tx) represent a topological space and {a} a singleton set, which is an open subset of X, {a}…

A:

Q: Problem 2. Let X be a countable set and let s e X. Consider the particular point topology Tp on X,…

A:

Q: * Let Z be with the discrete topology.11 If A = {1,3,5, 7,...}, then Aº %3D %3D A O N O R O

A:

Q: Theorem 3.15. Let (X,T) be a topological space, and let S be a collection of subsets of X. Then S is…

A: Given  X,τ is a topological space and S is a collection of subsets of X. Explanation let S is a…

Q: Question 5. In a topological space (X,7) a subset ACX is called regular open if A=int(cl(A)) "A= A".…

A:

Q: Let (X,7) be a topological space and ACX. Show that (n) A is open if and only if Int(A) = A. (b) A…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: (8) Show that the map f(z) = 1/z is a bijective holomorphic map from S? to itself, using the charts…

A: Note: A function is holomorphic if it is analytic everywhere except one point. Given: fz=1z, z≠0

Q: Theorem 6.15 (Heine-Borel Theorem). Let A be a subset of Rstd. Then A is compact if and only if A is…

A:

Q: Question 7. Let A=[0,1]U(1,2)ng) be a subset in the usual topological space where Q is the set of…

A:

Q: Theorem 2.28. Let A be a subset of a topological space X. Then Int(A), Bd(A), and Int(X – A) are…

A: Given that A is a subset of the topological space X. We have to prove that IntA, BdA and IntX-A are…

Q: on R. Question 3. Let r, and be two topologies on X such that Sr,. Construct a space on which a…

A: According to our guideline we can answer only first question only or three subparts  and rest can be…

Q: Question 5. In a topological space (X,t) a subset ACX is called regular open if A=int(cl(A)) "A A".…

A:

Q: Question 6. In a topological space (X.r) a subset Bof X is called a 7- open subset of X if…

A:

Q: Question 1. Let ACX. Show that 1= {UcX:AcU}U{®} is a topology on X.

A: For a set X having certain collection of subsets represented by τ will be a topology on X if it…

Q: Question 2 2.1) Let X = R and ( be a collection given by ( = {0}U{0cX:0°<}. (a) Show that ( forms a…

A:

Q: 6. Consider the horizonal strip 2 = {z+ iy : -1 < y < 1}. There exists a holomorphic function f on…

A:

Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

Q: Theorem 7.7. LetX be a space with a dense set D, and let Y be Hausdorff. Let f : X → Y and g : X → Y…

A:

Q: Question 2. space is the union of a finite number of closed balls radius e. Prove that a metric…

A:

Q: QUESTION 15 Let f. X Y be a continuous surjection between metric spaces. If X is compact then O AX…

A:

Q: Question 1. Let (X,r) be a topological space and let A be a subset of X. Determine whether the…

A:

Q: Question 4. Let f:X Y be a mapping from a topological space (X,7x) a topological space (Y,Ty). Prove…

A:

Q: QUESTION 13 A subset I of a metric space R with the usual metric is compact if and if only it is an…

A: In usual metric space a set is compact iff it is closed and bounded.

Q: Question 1. Given a metric space . (a) If 2, y E X and p(x, y) 0, prove that a y. (b) Prove that a…

A:

Q: Question 5. Let S be a subbase for a topology r on X and let A be a subset of X. Show that the class…

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

Q: Question 4. Let f:X→Y be a mapping from a topological space (X,7x) a topological space (Y,ry). Prove…

A:

Q: Question 1. Let X and Y be infinite sets. Let X be equipped with the discrete topology, and Y…

A: Given, X and Y be two infinite sets in topological space.  X is a discrete topology and Y is a…

Q: 2. Let (X, A, μ) be a measure space. If {E} CA and E₁ E₂ E3 C..., then prove that H(UE) = lim (En).

A: Let X,A,μ be a measure space. Let {En}n=1∞⊂A and E1⊆E2⊆....then, To prove: μ∪n=1∞En=limn→∞μ(En).

Q: Theorem [4.1.11] Let A and B be subsets of topological spaces X and Y respectively. Then Int(A × B)…

A:

Q: Question 2 Prove directly (i.e. from the definition of compactness) that if K is a com- pact subset…

A:

Q: Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least…

A: The given question is related with real analysis. We have to prove that a nonempty closed subset of…

Q: Question 6. In a topological space (X,7) a subset ACX is called regular open if A=int(cl(A)) "A= A".…

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

Q: Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each of X and Y is…

A:

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: Question 8. Let A=[0,1]U(1,2)ng) be a subset in the usual topological space where Q is the set of…

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

Q: Given, we have a finite purely non-atomic measure space (X, F, µ), prove that µ(F) = [0 µ(X)].

A: Given: Here, the given is finite purely non-atomic measure space (X,F,μ). To prove that μ(F)=0, μ(X)

Q: Question 2: Show that the set S = {rVT + V2:rE Q) is dense in IR. Explain and show your work.

A:

Q: Question 7. Let A=[0,1]U(1,2) ) be a subset in the usual topological space where Q is the set of…

A:

Q: Theorem [4.1.11] Let A and B be subsets of topological spaces X and Y respectively. Then Int(A × B)…

A:

Q: Question 6. In a topological space (X,t) a subset B of X is called a y-open subset of X if BC(B°UB).…

A:

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Show f is…

A:

Q: Question 2. Letr be the class of subsets of R consisting of R, ¢ and all open infinite intervals E,…

A: Given below the detailed solution

Q: 1. Let X C R"-1 be bounded. Prove that X considered as a subset of R" is a volume zero set.

A: Given, X⊆ℝn-1 That is, X is a subset of ℝ in terms of exponential n. So, let we consider the set,…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.