Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both infinite?

Q: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

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Q: Let K and L be two complete subsets of a metric space, then: * O KUL is complete but KNL is not…

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Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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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 (N, d) be the usual metric space, then the set A= {1,3} is O closed only O neither open nor…

A: ( N,d) be a usual metric space of natural number.  A={1,3} be a finite subset of N. Since A is…

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: a) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

Q: Prove that the quotient of a locally connected topological space is locally connected.

A: We have to prove that Let q:X→Y be a quotient map, X is a locally connected space. Show that xY is…

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: A completely regular space is also T1-space 1 True 2 False

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Q: 5. Prove that the property of a space having an isolated point is a topological property. 6. For any…

A: Sol

Q: Let X be a topological space

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Q: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: Every metric space is a hausdorff space .

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Q: A metric space is sequentially compact if and only if it is totally bounded and complete. prove it

A: We need to prove that any given metric space is sequentially compact if and only if it is totally…

Q: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: Let X and Y be two discrete spaces, then! X is never homeomorphic to Y X is homeomorphic to Y if and…

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Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If AC B,…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If ACB,…

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Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

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Q: Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only…

A: Given: X,d is a metric space, x∈X and A⊂ X is a non-empty set. To Prove: dx,A=0 if and only if every…

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

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Q: If (X.r,) and (X,r,) are T.-spaces, then (X.r,nr,) is also a T,-space. GOOD LUCK

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Q: let C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

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Q: Let [a, b] be an inter a space. Let ƒ:[a,b] → X

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Q: Let X and Y be compact spaces. Then X × Y is compact.

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Q: Let X and Y be two discrete spaces, then* O x is never homeomorphic to Y O X is homomorphic to Y if…

A: As two discrete spaces are homeomorphic iff they have same cardinality .

Q: Consider the spaceZ+with the finite complement topology. Is thisspace a Hausdorff space? Is this…

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Q: Prove that (R", II-|) is complete metric space.

A: This is a problem of Functional Analysis.

Q: Show that the closed ball Y in a metric space (X, d) is a closed set. Also, show that if (X, d) is…

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Q: Iwant to define these Poin (Defined) 1-) sobolve spaces 2-) Reflexive real Banach SPace

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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: 3: Prove that any reflexive normed space is complete.

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Q: Every closed subset of a connected metric space is connected. O True O False

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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: (a) A discrete space with more than one point is disconnected. (b) Any trivial space is connected…

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Q: Let N and N' be normed linear spaces with the

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

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Q: Let X and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the…

A: X and Y are two discrete spaces then X and Y   The topology is  discrete then for x∈X  is an open…

Q: Q7: Prove that ( IR", II-||) is complete metric space.

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Q: Show that a space X is Hausdorff if and only if the diagonal {(x, x)|x € X} is closed in X2.

A: We use basic topological definition to prove this.

Q: Every closed subset of a connected metric space is connected. True False

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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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Q: Let X be a discrete spaces then * X is never homeomorphic to R O X is homeomorphic to R if and only…

A: Fourth option is correct.