Consider the spaceZ+with the finite complement topology. Is thisspace a Hausdorff space? Is this space aT1space?

Q: Show that every second countable (A2-space) space is separable.

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Q: (a) Define the T1-space.

A: Define the T1 - space

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

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

Q: (2) Find all connected covering spaces of RP2 v RP².

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Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

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Q: A topological space is Hausdorff iff limits of all nets in it are unique-

A: This is a problem of Topology.

Q: A ° = A – JA

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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: Prove that any Space is complete. finite dimensional normed

A: Introduction: Completeness is a property of a topological space. When in a topological space, every…

Q: Every Hausdorff space is hereditarily Hausdorff.

A: We have to prove that every Hausdorff space is hereditarily Hausdorff. Consider a Hausdorff…

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

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Q: Let X be a topological space

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Q: Define a compact space.

A: The objective is to give the definition of a compact space.

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: 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: Prove the following: The countable product of metrizable spaces is a metrizable space.

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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: Any cover of a metric space is open.

A: Any cover of a metric space is open_____?_____

Q: If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff. The contrapositive of this…

A: Given : (1) (ℝ, T) with the topology induced by taxicab metric.              (2) (ℝ, T) with the…

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: Show that R is a bianch space.

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Q: Prove that any reflexive normed space is complete

A: A normed space X will be said to be complete if each Cauchy sequence in X is convergent in X.

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: If x is a limit point of a subset A of a metric space X, then sh

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Q: If A is 3 × 3 with rank A = 2, show that the dimension of the null space of A is 1.

A: Given: A is 3*3 matrix with rank A = 2

Q: Prove that the image of a continuous mapping of a compact set is compact.

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Q: / Let (X₁T) and (Y₁J) be top. Spaces and fix-x be a surjective and continuous map, then y is Compact…

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Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

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

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Q: The intersection of any countable family of open dense-sets in a locally compact space X is dense.

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

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Q: (b) Prove that the unit ball of a normed linear space is compact if and only if the normed linear…

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

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

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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: Every inner product space is a metric space O True O False

A: TRUE

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: 2. Let (X, 1) be any topological space and (X, Tf) be the finite complement topological space. Show…

A: Let (X, τ1) be any topological space and (X, τf ) be the finite complement topological space.Show…

Q: Let X be an inner product space. Then show that the projection theorem holds if and only if Riesz…

A: Given that: X=Inner product space. The objective is to show that the projection theorem holds if and…

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: (a) Prove that the metric space (Z,) is complete.

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

Q: For a normed space X, prove that the dual of X is separable implies that X is separable. Is the…

A: NOTE: According to bartleby we have to answer only first question please upload the question…

Q: Show that the closure of a connected space is connected.

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.