Let (X.d) be mnrtric space and X is closed then X is O complete Ono decsion Onot complete

Q: c. Suppose that G,H < Sym(Q). Show that the corresponding spaces (2,G) and (Q,H) are equivalent if…

A: Suppose that G, H ≤ Sym (Ω). Let G, H be a multiplicative group and let Ω be a set. An action of G,…

Q: Let E be a subset of a space X. show titat cl(E) = EUd(E). %3D

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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: Let (R, d) be usual metric space, then Ris ot separable True False

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Q: Let R be the usual space * In R, if A = [-4,0) U N, then A' = ON O [-4,0] U N O [-4,0) O [-4,0]

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Q: Let (R. d) be discrete metric space then R is not complete O True O Faise

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: Provethat amappi'ng from a metric SpaceXto metri'c space X is con tinuok ifand only f the inverse…

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

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Q: Olet se be u lecally compact topological space. Prove that that GL2) in a cEalyehra (ie. prove (a)…

A: Given:- Ω- Locally compact topological space. We have to prove that C0(Ω) is a C*-algebra means,…

Q: be nonempty Subsets of Space X. Then Prove Let A and BJA inner product an TT 10) A CAL TT %3D (C) A…

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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: Prove the following: Suppose that {X; : i € I} and {Y; : i E I} are indexed families of topological…

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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: Let X = R and s be a collection given by ( = {0}U{0cX:0° <x}. (a) Show that S forms a topological…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c)…

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Q: Let X and Y be normed spaces. Then show that the following hold : (i) If Y is a Banach space, then…

A: We consider an arbitrary Cauchy sequence (Tn)n≥1 in BL(X, Y) and show that (Tn)n≥1 converges to an…

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: al let (xt) betopology space and BCY then B is closed in y iff there is closed set CinX st B-COY

A: Let (X,Γ) be topology space and B⊂Y To Prove: B is closed in Y iff there is a closed set C in X s.t…

Q: rove that spaces Ip for os p<l If for pe(0, 0) {0} and complet. are

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Q: Let X = Rand def ine ||.|| on X as ||x|| = 1 %3D then X is not normed linear space True O False O

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Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

Q: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

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Q: Q3 Assume that (M,d) is a Compact metric Space. Show that, f: (M,d) → (Y, 2) is Continous bjective…

A: f:(X, d) to (Y, d) is said to be homeomorphism if  1) f is continuous.  2) f is bijective.  3) f…

Q: The following statements about a topological sapce are equivalent: (a) X is a Hausdorff space; (b)…

A: Given: The statements: a) X is a Hausdorff space; b)  The diagonal D=x, x: x∈X is closed in X×X. To…

Q: Let X= {A ⊆ ℝ| 1 is not in A and ℝ \ A is finite}. Prove that ℝ is a compact Hausdorff space with…

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Q: let X be a firite dimensional notm Space Then Prove that is compat.

A: As X is a finite dimensional norm space, this means that, dimX is finite.

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

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Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

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

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Q: let (2, m) be a meesurable space. then 1f f9: x LJare measural ble function, then -> ) f43 is mea…

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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. biso (2) If…

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

Q: 4. Let (x, d) be a metric space and ACX be nonempty set. Define f:X → IR, fix)= inf dex,a). aE A…

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Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

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Q: Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

A: Given:  (Y,d) is a metric space, and Y is totally bounded To Prove Y is bounded Definition: Let…

Q: let a-Rbe ametric sPacesConsir he following subsed ofP @ Show that Dra De are comPact Setsornot of D…

A: Question from real analysis.

Q: IfS is a subset of an innerproduct space V(F) then show that Scs=(0}. %3D

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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 (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: Let X=R and T = {]a, +oo[/a R}U{o,R}, then (X, 7) is a regular space. True False

A: This is a problem of topology.

Q: The following statements about a topological sapce are equivalent: (a) X is a Hausdorff space; (b)…

A: We have to prove that the following statements are equivalent : (a) X is Hausdorff space (b) The…

Q: Show that the set of all real number constitutes incomplete metric space an we choose doxoy) = Itan'…

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Q: Prove disprove that (R.Tco-finite) is Te-space or

A: To Prove - Prove or Disprove that ℝ, τco-finite  is T2 - space.       Definition used - T2 - space…

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

A: TRUE

Q: tet is cosinte toploy on set R show that (R, t) is O NOVmal space- Regualra space

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Q: ii). what { disie [4NI} to fobrm vedtor space ? set vectas a bauis for en N-dimeneional alte the…

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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 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: 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…