3. Let (X, d) be a metric space and x1, x2, ... , Xn, ... a sequenc

Q: 3.2 Prove that in any metric space (S, d) every closed ball S,xo is a closed set.

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r>0…

Q: 4:: what the Null Space of is T;: R3 R³( T,x= xx a) R, R (T = X.Q

A: As per guidelines, we will solve first three sub-question: We have to find the null space of  1.…

Q: (2.3.15. Let X be a metric space. (a) Prove that SCX is bounded if and only if diam(S) < 0o.

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

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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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: Let (X,d) be metric space and X is unbounded, then d (z. y) . y EX, is 1+d(r,y) di de fined as dy =…

A: Metric space

Q: Let (R. d) be discrete metric space then R is not complete O True O Faise

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Q: - Let (X, t) be a topological space and A, BCX, then 1- A° 2 A- b(A) 2- A° # A- b(A) 3- A° = A –…

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Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

Q: Show that an infinite intersection of closed sets F, k = 1, 2, 3, . .., in a metric space (S, d) is…

A: first we will proof that an infinite union of Open sets in metric space ( S, d) is open using the…

Q: If there exists open ball B (a, r) such that B (a,r) nG = Ø then a is not limit point of G True…

A: TRUE

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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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: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and ВС (Х, М) :%3D…

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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: 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: Construct examples to show that if X and Y are topological spaces with G acting on them such that…

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Q: The set [0, 1] with the discrete metric is compact. O True O False

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Q: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

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Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M);…

A: Given as X,T be a topological space,M,d be a complete metric space and BCX,M=f∈CX,M;fX is bounded…

Q: Prove that: 1. (R, |. |) is a complete metric space. 2. (Q.I-1) is not complete metric space.

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

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

Q: 3.4 Give the definition of a separable metric space (S, d). 3.5 Show that the metric space (R, d),…

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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: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

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Q: Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

A: Given:  (Y,d) is a metric space, and Y is totally bounded To Prove Y is bounded Definition: Let…

Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

Q: (3) In R² with the standard topology: {(x, y) | x² + y² {(x, y) | x² + y? > 1}. = 1}, {(x, y) | x² +…

A: In a topological space R2with standard topology. A  subset O is open if at each point x in O there…

Q: 4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed.

A: Let (X,d) be a metric space, with x∈X , ε&gt;0, and E={y∈X:d(x,y)≤ε} To show: E is closed. That is…

Q: Example 3.16. In R with the Euclidean metric, the set [0, 1] is compact. However, note that with the…

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Q: 3.27.2. Let X be a metric space with metric d; let ACX be nonempty. (a) Show that d(x, A) = 0 iff x…

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Q: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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Q: Let X=R and T = {]a, +oo[/a R}U{o,R}, then (X, 7) is a regular space. True False

A: This is a problem of topology.

Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

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Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

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

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Q: (2) Let (X,r)be a topological space, If for every A open set, and x E A", 3 Ux and Vy open sets,…

A: A topological space X is Hausdorff if for each pair of distinct points p and q in X , there are…

Q: Let (R, d) be the metric space with d(x, y) = /[x – y[. Then the open ball O [-3, 5] ]-3, 5[ ]-1, 3[…

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Q: (c) If X={1, 2, 3}. T= {4, X, {1}, {1,2}, {3}, {1,3}}. JUSTIFY whether the topological space (X, t)…

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

A: Given set  S= {x€R : x2 -4&lt;0}

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

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Q: c) Let (M, d) be a metric space and let S,T≤ M. Prove that SOT = ST. But SOT SOT.

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Q: 8- Suppose that X,Y be a normed spaces then the function f: X → Y is continuous at a point xo E X if…

A: True

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

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Q: 1/ For any. topological space (X, T), if T has only three elements then (X, T) is connected.

A: Given: Topological space X, T and T has 3 elements  To prove that X, T is connected when T has 3…

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: 2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined by ρ(x, y) = d(x, y) 1 +…

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