5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. ореn b. closed c. clopen d. is neither open nor closed

Q: Let M = {0,1} and d be the discrete metric, then d is not complete %3D O True O False

A: Discrete metric space.

Q: 2. Let (S, d) be a discrete metric space and p E S. Identify S i(P), S1 (p), S2(p).

A: We shall provide answers for first question only as per the answering guidelines. For other…

Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

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Q: 21. Let A and B be two on-empty bounded sets of positive embers and let С%3D {ху: х€ А and y € B}.…

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Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Then X" is homeomorphic…

A: Proof-) The Cantor set itself is a cantor space. But the canonical example of a cantor space is the…

Q: Suppose that l is a line. Let H be one of the two half-planes bounded by l. Give a proof that l U H…

A: Answer is mentioned below

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

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Q: (b) Prove that the set S = {x EQ: √2<x<√3} is closed and bounded in (Q.1-1), but not compact in…

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Q: Let (X,T) be a discrete space, then Va e X, the set {a} is а. орen b. closed c. clopen d. is neither…

A: Given problem is :

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

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Q: O Show that if Si and S2 are convex sets in R™X", then so is t mxn S = {(x, y1 + y2) | x € R", y1,…

A: Convex set: , If a and b are points in a vector space S. Then the vector space is called convex set…

Q: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

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

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Q: Let (R, d) be the metric space with d(x,y) = |x – yl. Then the open ball B(1,2) is None of these O…

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Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

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

Q: Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c)…

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Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: let (x, m) be a measurable space. then ¥8:% [, o] are measural ble funetion, then ) I +9 is mea…

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Q: Suppose metric space X is not path-wise connected then X is not connected. O True O False

A: Question 9 Answer: as a path connected space of employees connected but converse is not true. So I…

Q: 5. Let (X,r) and (Y,A) be two topological spaces. Prove that a function f:(X.r)(Y.A) is closed if…

A: Since you have posted multiple question according to company rule we are supposed to give solution…

Q: Let X be an uncountable set and let oo EX be fixed. Let TX = {U E P(X); 0$U or X- U is finite} Let Y…

A: Given: Let us consider the given X to be unaccountable and let ∞∈X be fixed.

Q: Make the definition of tupological non-empty set and * E X. Inves tigale whether Space Let X be…

A:

Q: Suppose (X,S,u) is a measure space and f,h: X → F are S-measurable Prove that I|fh||, < ||f||, ||4|a…

A: Given X,S,μ is a measure space and the functions f,h:X→F are S-measurable. Suppose 1p+1q=1r,…

Q: Let X = R, and Let M = [0, 3] and d be the discrete metric, then M is

A: M=[0,3] is compact ,since it is closed and bounded in real R.

Q: Suppose that f:[a,b] R is bounded, P is parition of [a,b] and Q is refinement ofP, then: L(f;P) <…

A: Answer

Q: 4. Let (x, d) be a metric space and ACX be nonempty set. Define f:X → IR, fix)= inf dex,a). aE A…

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Q: 8- Let X, Y are metric spaces and f: X → Y .If for any A C X, f(A) C f(A) then f is continuous.

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Q: Let X = R, and d be the discrete metric, and A = [3, 6) Show that if A is compact or not

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Q: Let (N, d) be the discrete metric space, then the set A = {2,3, 4} is closed but not open open but…

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Q: The set S=xER:x²-4<0} with the usual metric is . O A. Compact. B. connected. C. Not connected D.…

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Q: 2. Explain why there does not exist a measure Space (X,S, M) with the pro porty %3D

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Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+…

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

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Q: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. орen b. closed c. clopen d. is…

A: Our objective is to choose the correct choice

Q: Let f : (X,d) - (Y, d1) becontinuous fumction, then for any closed set G in X, f(G) is not necessary…

A: False

Q: 1- Suppose that ,p) is a meric space, an that E CX. Then the interior E is open (m X) and the…

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

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. O If f…

A: Given: (X,T) &amp; (Y,T1) are two topological spaces &amp; f is a continuous mapping of X into Y.…

Q: Please solve this linear algebra question. Thank you.

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Q: (b) Given two metric spaces X,p>, and a function f : X→ Y that is uniformly continuous on SC X. If…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

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Q: 4.Describe the largest set S on which it is correct to say that f is continuous S (x,y) = In(1–x² –…

A: Given problem:- Describe the largest set S on which it is correct to say that f is continuous C.…

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: 5. Let (X,7) be a discrete space, then Va E X, the set (a} is a. open b. closed c. clopen d. is…

A: As per our guidelines we are suppose to answer onlyone question, kindly post the other questions as…

Q: Let N be with the discrete topology If A=(1,3,5..), then Bd(A)=

A:

Q: Let X= (3, 5] u {6,7,...} and A= {5,6,7} and metric defined on X as d(x,y) = |x - y|

A: Note: As per bartleby instruction when more then three questions for subpart is given only three…

Q: Let X = {a, b, c, d, e} and let T ={X, Ø, {a}, {c,d}, {a,c,d), {b,c,d,e}), therm %3D O (X,T) is…

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Q: Let (X, Tx), (Y, Ty) represents topological spaces such that X = {a,b,c,d}, Tx = {X. Ø, {a}, {a, b},…

A: CONTINUOUS FUNCTION: If the inverse image of every open subset of Y is open in X, then the function…

Q: (b) Describe graphically the set A={(x,y):d[(x,y), (0,0)]<3}for the following metric space on R²…

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Q: Let X and Y be two discrete spaces, then *

A: Note: Since you have posted multiple questions we are only allowed to solve the first one. Thankyou…