Q1: (a) If (M, d) be a metric space and A M. Prove that A is the largest open set in M that contained in A. then give an example to show that (AUB)* # A U Bº. (b) let (M, d) be a metric space and let ACM. Then (ANB) ≤ANB

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: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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Q: Elaborate on the role of duality in topological spaces.

A: Introduction:Topology is a branch of mathematics that studies the properties of spaces and the…

Q: Prave R is banach Space with explanation.

A: Banach space: A complete normed linear space is called Banach space. Given:  To show:  is Banach…

Q: Let R be the set of real numbers and d: R x R → R be defined as follows. d(x, y) = 3|x8 – y®|. %3D…

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Q: (E, Q4: state and prove L is a normed space ? e a ne. Vila a convex set.

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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: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: Which of the following statements is not true? (i) In any metric space (X,d) every closed ball is a…

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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: ctors - Z. S. Madhi

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Q: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

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Q: (a) Prove that the metric space (2,11) is complete,

A: Complete Metric Spaces:- A Metric Space (X,d) is said to be complete iff every Cauchy sequence in X…

Q: 4={(-21.r):reR} II) B=(2r.-31): te R} III) C={(2r+3,-41) re R} I)

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Q: Define the Cartesian product of sets X and Y? How is this Cartesian product denoted?

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Q: Q1) Prove or Disprove: Let L(F) be a linear space and A,B are two convex subsets of L. Then, AXB is…

A: The problem is related with convex set. Given that,                 LF  is a linear space  and A, B…

Q: Give an example of 3 vectors spaces that are not Rn. Explicitly state the definition of addition and…

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Q: Give an example of a norm space.

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Q: Determine whether the statement is true or false.   b) The set of all ordered triples (x, y, z) of…

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Q: Determine whether each product is an inner product space in R3. If not, explain why. 1) = u1v1 +…

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Q: Quertion:- Show that R is a Banach Space.

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Q: (a) Show that every metric space is T2-space.

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Q: ACX A + Ø and A + X. Let B = AU fr(A). .show that for the topology T = {X,Ø, A, B} it happens that…

A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology.  Here B=A∪frA To prove for…

Q: Show R³ is a banach space.

A: Banach space: A complete normed linear space is called Banach space. Given: To show:  is a Banach…

Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: Q6: (a) In any metric space (S, d), prove that every compact subset T is complete. (b) Prove that…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

Q: a) Find a space that is a regular but not Hausdorff. Justify your claim.

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

Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…

Q: Show that if X and Y Hausdorff spaces, then so is the product space X × Y. are

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