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

Q: If τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normal space…

A: τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normalspace or…

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

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

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

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Q: Show that R is a Banach space.

A: Banach space: A complete normed linear space is called Banach space. Given: ℝ3=x(1, x(2), x(3)) :…

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,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

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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: Let (X, d) be a metric space and let A ⊆ X be complete. Show that A is closed.

A: Given that X,d be a metric space and A⊆X be complete. The objective is to show that A is closed.

Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

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Q: Prove that if f is a continuous mapping of a compact metric space, X, into a metric space Y, then…

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

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Q: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

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Q: Let (X.d) be a metric space, x ∈ X, epsilon > 0, and E = { y∈ X: d(x,y) ≤ epsilon}. Show that E is…

A: A set E is closed in X if it contains all of its limit points.

Q: Let X be a topological space

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Q: let and be nonempty metric Space x E X. Ihen A B Sebsets of and let d(x, A), dlA, B) and dlA) are…

A: As per bartleby guidelines for more than three subparts in a question only first three should be…

Q: Every metric space is a hausdorff space .

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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: 1. (a) Define what a linear subset of a projective space is.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Consider IR as a metric space.with metric dux, y) = 1x-y1. Show that the open interval C0, 1) is an…

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Q: Every T3 space is regular space. True False

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

Q: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

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Q: Let (X, T) be a topological space and A, B be a subsets of X then: اختر احدى الدجابات int(A - B)=…

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

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Q: Question 10. Prove or disprove. Any infinite subset A of a discrete topological space (X,r) is…

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Q: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

A: Let X={a,b,c} be a set. To show that T={X,ϕ,{b},{a,b}} be a topological space on the set X.

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 X and Y be compact spaces. Then X × Y is compact.

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

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Q: A of a metric space (x, d.)

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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: (d) Every continuous map on any metric space is an open map.

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Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: Given that, X is a discrete space. To be homeomorphic X has to be of same cardinality as ℝ and so X…

Q: Prove that Let X be a metric space such that X E1 U E2. If E1, E2 are complete, then X is complete.

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Q: If X is an inner product space and A = {0}, then A = %3D %3D O X O A

A: A⊥ = X

Q: 1. Show that if & is a function from a nonempty compact metric space X to itself such that d(@(x),…

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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: If X is a metric space and Y C X, then Y is a metric space.

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Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: Show that (C(X),|| · ||∞,x) is a metric space.

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Q: Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only…

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Q: Let (X7) be a topological Space and Show that A is open and closed m (xx) iff %3D

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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: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

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: If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is…

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

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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.

Q: A° U B° c (A U B)

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