Exercise 3.55. Show that every wedge sum of Hausdorff spaces is Hausdorff.

Q: Prove that any contraction between metric spaces has at most one fixed point

A: According to the given information, it is required to prove that any contraction between metric…

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 X be a locally compact Hausdorff space which is not compact; Let X* be one-point compactificatin…

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Q: LxercisE Let (X,2,4) be a mensure spsce. Let 1 2- (0, 4oo and pn : 2(0,+o0] be two menaures. Prove…

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Q: Prove there exists a continuous surjection from the unit interval to the 2-sphere

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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: every metric space is normed space? give an example to show tha

A: We have to show that every metric  space  may not be a  normed  space.

Q: Don't use chat bot     Show that a compact Hausdorff is metrizable if it is locally metrizable.

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Q: Determine if the following statements are true or false. Briefly explain why. a) A quotient of a…

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Q: Every metric space is a hausdorff space .

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Q: cise 2 Let X be a topological space. - Show that X itself is a Baire space. T

A: Given: X is a topological space. To Prove: We have to show that the topological space X itself is a…

Q: Find an example of a topological space that is seperable and first countable but not second…

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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: 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: Show that: All completions of a metric space are isometric.

A: According to the given information, it is required to show that all the completions of a metric…

Q: Every T3 space is regular space. True False

A: Since a regular T1 space is called a T3 space.

Q: * Use the Grom Schmidt process for thi two_ dmensional space: 2 3. 4 2

A: Gram Schmidt process is given by: uk→=vk→−∑j=1k−1uj→⋅vk→|uj→|uj→   v1→=201, v2→=3-51, v3→=042,…

Q: Let X and Y be two topological spaces. X is connected and Y is disconnected are they homotopy…

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Q: 2.8.15) Prove that a compact metric space is both complete and separable.

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Q: If S be a n-dimen Sional normed Space, then prove that its dual space is al so n- dimen Sional .

A: A normed space E is called a Banach space if it is complete, i.e., if every Cauchy sequence is…

Q: Prove or Disprove the following: a) The product Space of two Connected spaces must be connected.…

A: As there are two proofs asked, I will solve only the first one. Let M1 and M2 be connected spaces.…

Q: Use F.Riesz Lemma to show that the closed unit ball in an infinite dimensional Banach space is not…

A: Let us prove above result

Q: Prove or give counter example an infinite product of discrete spaces may not be discrete

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Q: True False An open interval is not homeomorphic to either a closed interval or a half open interval.…

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

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Q: Approximation Theory Q1\ Answer true or false 1) A normed space is a non-empty set with norm. 4) The…

A: According to our guidelines we can answer only three sub-parts and rest can be reposted.

Q: Prove the interval [0, infinity) is closed but not bounded

A: Consider the provided question, We have to prove that the interval [0, ∞) is closed but not bounded.…

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

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Q: 6. A space X has the property of being separable if it contains a countable dense set (its closure…

A: Solution:      We have to show that, the property of being separable is a topological property. Now,…

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

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

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Q: Define what it means for a topological space to be completely regular,

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Q: Show that all open balls are bounded subsets of a metric space.

A: Solution: Let M=X,d be a metric space. For an arbitrary point a∈X, the open ball Bεa where ε>0 is…