Construct examples to show that if X and Y are topological spaces with G acting on them such that X/G = Y/G then X and Y are not necessarily homeomorphic.

Q: Let A and B be topological spaces and endow A x B and B x A with respective product topologies. Show…

A: Homeomorphism: Let X and Y be topological spaces; let f:X→Y be a bijection. If both f and f-1:Y→X…

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

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Q: Theorem 3.7: A subset A ofa metric space (X, d) is closed if and only if /.contains all its limit…

A: We have to prove that A is Closed if and only if A contains all its limit points. Note : In proof i…

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: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

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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: show that any infinite discrete topological space is not compact

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Q: 6. Show that a compact metric space X is locally compact.

A: The given problem is related with metric space. We have to show that a compact metric space X is…

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: 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: 2) let X and y be metric spaces and T. XY Y a ContinUOus mapping Then the image of a Compact subset…

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Q: Let X be a locally compact Hausdorf f space

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Q: Let X and U be Banach spaces, let C: X→ U be a compact linear map, and let xnx in X. Show that Can →…

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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: Prove that coo cannot be a Banach space under any norm.

A: To prove: c00 cannot be a Banach space under any norm.

Q: Let X be a topological space

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Q: Theorem 3.7: A subset A of a metric space (X, d) is closed if and only if/.contains all its limit…

A: A subset A of a metric space (X, d) is closed iff A contains all its limit points.

Q: Prove that an affine map is continuous if and only if its linear part is continuous

A: If T:A→Rk is the affine map, where A is the affine span of an affine independent subset of Rn. Let…

Q: Let X and Y be two locally compact spaces, then X*Y is locally compact. (Recall that: X is said to…

A: This is a problem of topology.

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 topological spaces. X is connected and Y is disconnected are they homotopy…

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Q: Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point.

A: We have given that, X is Compact space.

Q: Prove that every finite T,- space has the discrete topology.

A: T=(X, τ)  (X, τ) is a T1 space if and only if all points of X are closed in T.   Let X≠∅ be a set.…

Q: Let X and Y be topological spaces and fix some yo E Y. Show that the map f: X → X x Y defined by f…

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Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

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

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Q: Show that any infinite subset A of a discrete topological space X is not compact.

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Q: Show that if a Banach space X is non-trivial, X + {0}, 3). then X' # {0}.

A: According to the given information, it is required to show that:

Q: Let X be a finite dimensional normed space (a) and A :X→X is a linear map. Show that A is compact.…

A: Given: X is a finite dimensional normed space and A : X → X is a linear map. We have to show that A…

Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: Let (X,T) be a topological space such that every subset of X is closed, then a. (X,T) is an…

A: (X,T) is a topological space such that every subset of X is closed

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

A: This is a problem of Functional Analysis.

Q: a) Consider the set X = {1,2,3} with the topology t = Show that (X, t) is a topological space. (0,X,…

A: As per our company guidelines we are supposed to answer only one asked question. Kindly repost…

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: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

Q: Suppose U and v are conformally equivalent. Prove that if U is simply connected, then so is V.…

A: Given that U and V are conformally equivalent that is there exist a mapping f from U to V such that…

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

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Q: Theorem 7.32. Let X and Y be topological spaces. The projection maps tx,Ty on X×Y are continuous,…

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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: a function f is continuous at x0 element of X if and only if f is both lower semi continuous and…

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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: Let (X,τ) is a topological space and A ⊆ X. If all subsets of A are closed in X, then set A cannot…

A: Given that X,τ is a topological space. Let's prove this theorem by contradiction. So  X,τ is a…

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: 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: Let X be ametric space. If A and B are each dense open subsets of X then AOB is dense open subset of…

A: First we have shown the intersection is open. Then we use the property that if a set intersect every…

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

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