Let X be a metric space. Let A and B be separated sets in X. Show either E is a subset of A or E is a subset of B where E is a connected subset of A U B.

Q: Is the euclidean space R³ is a complete metric space

A: We have to check whether euclidean space R³ is a complete metric space or not .

Q: Prove that in a metric space (S,d) every closed ball Sr[Xo] is a closed set

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Q: • A is open in X. • A = Int A • A contains none of its boundary points. • Every point of A has a…

A: As per the company rule, we are supposed to solve one problem. Therefore kindly repost the remaining…

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

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Q: Determine it this new def ot O wald make Va Vector Space

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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: Given a metric space M with metric d, verify that any e-ball is an open set.

A: Given That : Metric space M with metric d . To Verify : Any ε-ball is open set.

Q: Every metric space is a hausdorff space .

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Q: Let the definition of disconnected be the definition in the picture attached. Show that this…

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Q: Let

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Q: B. Given a metric space M with metric d, verify that any ε-ball is an open set. (Please If possible…

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Q: Consider the following topological spaces: Space X: A closed unit interval [0, 1] with the usual…

A: Two topological spaces X and Y are said to be homeomorphic if there exists a continuous bijection f:…

Q: Say whether (?, ?) is a topological space. Briefly explain your answer i. ? = R, and ? =…

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Q: Q. Let S = {(x, y), x² + 4y2 = 4} CR2, then S - {point} is homeomorphic to – y? = 4} a. {(x, y), 4x2…

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

Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

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Q: Let R be the set of all real numbers with the euclidean topology. Define the set A = {(r, y) ER x R…

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Q: B) Let X = R be a metric space and consider the following subsets of X, %3D 1 1 S = (0,1), S2 %3D…

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Q: Exercise 1. Let X be a metric space. The subsets of X which are arbitrary (finite or infinite)…

A: What is Open Set: Open sets in a metric space are those that, for every point P, include all other…

Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: A, B be subsets of a metric space. Show that A U B = B and that A B. Give an example to show that An…

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Q: Given a metric space M with metric d, verify that any e-ball is an open set.

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Q: Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: Let H = Span{1+x,1+x², 3+x +2x², 2 + x + x²}. what is dimH?

A: Solution

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 R be the set of all real numbers with the euclidean topology. Define the set A = {(r, y) ER x R…

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Q: Let 0 be the G-space and t be a fixed element of G. The pair (0, p), where 0: N → N and p: G → G are…

A: Let σ be a G-space. That is, there exists a continuous map from G×σ→σ. Now define two maps θ,φ as…

Q: Q. Let A = {(x, y), y = x2, x E Z, y E R}, Then A is a. connected and pathwise connected space. b.…

A: Neither connect nor path connect

Q: For any metric space (X, d) and x EX, the singleton set (x) is closed. Select one: OTrue O False

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Q: Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.

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Q: Prove or Disprove. Equivalent norms on X give the same norm of each x belongs to X.

A: The given statement is true. So we have to prove that : Equivalent norms on X give the same norm of…