Let (X, M, u) be a measure space wE

Q: Q2) Let (A,T)be topological spaces; i=1,2,3. Then A, x A2 x Az is homomorphic to (A, x A2) x A3 I

A: Definition : Let X and Y be spaces, and let B be the collection of sets U × V , where U ⊆ X and V ⊆…

Q: Suppose A and B be subsets of metric space. Show that if A is compact and B is closed then A + B is…

A: Given A and B are subsets of a metric space. and also Given A is compact , and B is closed. We have…

Q: Show that every metric space is an Hausdorff space.

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Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

Q: Let E be a subset of a space X. show titat cl(E) = EUd(E). %3D

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Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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Q: Let (X.d) be mnrtric space and X is closed then X is O complete Ono decsion Onot complete

A: A metric space X is said to be complete if every cauchy sequence in X is convergent. That is, every…

Q: Let R be the set of real numbers and d: R x R → R be defined as follows. d(x, y) = 5|x10 - y1º|.…

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Q: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: Let X be a topological space

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Q: Define metric space and define its four properties

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Q: State true or false with brief justification: L1[a,b] is a reflexive space

A: We use the Theorem: Let X be a separable normed space. If the dual space X* of X is non-separable,…

Q: This answer is incorrect, could you please attempt it again?

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Q: 10. a b C d a + 3b = c b+c+a=d

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Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

Q: Show that the spaces (R,U) and (R,S) are Hausdorff.

A: Let a, b be any two distinct points of R so that a ≠ b

Q: Let ||u|| = 2, ||v|| = 3, (u, v) = 2+i. Compute ||u+v||², ||u-v||², (u + v,u - iv), (u + 3iv, 4iu).

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Q: Q4. Consider the topological space (X,T), where X = {a, b, c, d) and T = {o, X, {a}, {b}, {c},…

A: Given: Q4. a)  X, T is a topological space, where X=a, b, c, d and T=∅, X, a, b, c, {a, b}, {a, c},…

Q: a soft drink can is a: A)can be both discrete and continuous B)discrete variable C) none…

A: A continuous variable is a variable that can take any specific value between two finite variables.

Q: Let V = R³ and W = {(a, b, c) E V | a² + b² = c?}. Is W a sub- space of V? If so, what is its…

A: As per the question we are given the three dimensional vector space V = ℝ3 and a subset W is defined…

Q: Let A = {AC R| A is countable} and A2 = {A C RI A° is countable} which is a measurable space? (IR,…

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Q: 1- In a metric space (X, d), if A and A2 are open subsets of X then A1 U A2 is open in X.

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Q: Let (IR, I) be usual 2 2 IR ² X IR² = Cx C space. Prove

A: Solution:  The  given equation f:ℝ2→ℂ   defined  by  f(a,b)=(a+ib), a,b∈ℝ2. Then,a,b∈ℝ2, and…

Q: Suppose A is a closed, non-empty subset of a metric space (X, d) and p a point of X which is not in…

A: Given:  A is a closed and non-empty set of a metric space (X,d) and p∈X , p∉A. To Prove: inf{d(p,a)…

Q: Q4. Simplify the following expression: 3-5 x 39 Click on image to enlarge if needed 12 81 243 324

A: To evaluate :- 3-5 * 39

Q: A⊂B, prove that if B is bounded, then A is also bounded. Is the converse true?(

A: Let A⊂B, where B is bounded. Then there exist two real numbers k and K such that k ≤ b ≤ K for all b…

Q: ovide an illustration of the statement "Every normal space need not be a ular space."

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Q: Let (R, d) be discrete metric space and A = N then A is complete and compact neither complete nor…

A: Given that (ℝ,d) is a discrete matric space and A=ℕ. then     d(x,y)=0 if x=y1  if x≠y A matric…

Q: Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

A: Let A be a closed bounded subset of a metric space (X, d). Is A acompact subset? Explain why.