If there exists open ball B (a, r) such that B (a, r) nG=Ø then a is not limit point of G

Q: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

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

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Q: If there exists open ball B (a, r) such that B (a, r) NG= Ø then a is not limit point of G True O…

A: TRUE

Q: Let R be bounded by the hexagon with vertices at the points (2,0), (3,1), (3,5), (2,6), (1,5),…

A:

Q: Let (X,d) be metric space and X is unbounded, then d (z. y) . y EX, is 1+d(r,y) di de fined as dy =…

A: Metric space

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: We define the included point topology by Tp={UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R if

A: In order for a set to be dense, it is required to be equipped in Tp and also as in the set 5 is not…

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: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded. %3D

A:

Q: If there exists open ball B (a, r) such that B (a,r) nG = Ø then a is not limit point of G True…

A: TRUE

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

A:

Q: The space X = {0,1} with topology T = {0, {1}, {0, 1}}) is To but not T Select one: O True O False

A: Discrete spaces are always T0 , So X is T0 .  A space X is a T1 space if and only if  for each x∈X,…

Q: Let (X, T) be a topological space and A,B CX- ConstYuct the general Yelation between Fr (IntlA) and…

A: For any subset B of a topological space X , Frontier or boundary of B is defined as  Fr(B) = CL(B) -…

Q: Give the set of limit points A0 of a singleton A = {(5, 2)} on the plane R 2 with the discrete…

A:

Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: Let (R. d) be discrete metric space, thend can not induces a norm on X O True O Fahe

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Date. AC) Lat normed linear space X, Y= spomE and aex only if fla) = 0 wheneres f EX' omd f30…

A: To show the necessary part- To show that for each f∈X' whenever f vanishes on E, i.e.if f=0 then…

Q: 9.A. Show directly from the definition (i.e., without using the Heine-Borel Theorem) that the open…

A: We are working in a metric space so compactness is same as sequentially compact. Consider, the…

Q: We define the included point topology by Tp={ UcR;U=ø or peU}. Let A = [3,5[, then A is dense in R…

A: In order to be dense in R, R has to be equipped in the Tp. Also as in the interval, 5 is not…

Q: Show that the maximum norm on C[a, b] is not induced by an inner product

A: to show that the maximum norm on Ca,b is not induced by an inner product . as we know , a norm X,·…

Q: (b) Given two metric spaces X,p>, and a function f : X→ Y that is uniformly continuous on SC X. If…

A:

Q: 12 Let (RTR) is topological Space Such that TR-243 U2GSR=R-G-G is finste} Find buz), Z², Z. z' then…

A: Here given ℝ, Tℝ is a topological space such that Tℝ=ϕ∪G⊂ℝ: ℝ-G=Gc is finite This topology is known…

Q: A. Let H be the set of all points (x, y) in R? such that x2 + 3y2 = 12. Show that H is a closed…

A:

Q: Show that the inner product (,) on an inner product space X is a bounded sesquilinear form h. What…

A: Given that X is an inner product space with an inner product ·,· defined on it. We know that a map…

Q: A. Let H be the set of all points (x, y) in R2 such that x2 + 3y2 = 12. Show that H is a closed…

A:

Q: ht x and y be normed spaces. Show that a Lincar operator T:Xy is bounded If and only T maps bounded…

A:

Q: 28. Bounded by the planes z = x, y = x, x + y = 2, and z

A: Solution:-

Q: Show that the closed ball Y in a metsic SPace (X,d)is a closed set. Also, show that it (x,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. also show that if (X,d) is…

Q: Show that every closed ball in R" is an n-dimensional manifold with Assuming the theorem on the…

A:

Q: th atlas V (so that D

A: Given: manifold (M, A) To show: V (so that D(V) = A)

Q: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

A:

Q: Let x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit…

A: Interior Point: Let A be a subset of metric space X, d. A point x∈A is called an interior point of A…

Q: Show that the closed ball Y in a metric space (x,d) is a closed set .Also show that if (x,d) is…

A:

Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

A:

Q: Let R be bounded by the hexagon with vertices at the points (1,0), (2,1), (2,5), (1,6), (0,5),…

A:

Q: Show that it A is subset of X any with the discre te metric then %3D

A:

Q: Let (X, t) is topological space, ACX, (X,d) is a metric space. Prove that: f: (X,d) -> (IR, Ts),…

A: Let ( X, d ) be a metric space and R with the standard topology. Then, we define a function  f(x) =…

Q: *let (Xn) a tn IR (1, 1, 1,) be sequence and d be the Usuat metric, then (X) is A-Converaent. B…

A:

Q: Show that, with z = Rw, for some bounded w, || z(jw)|| w llw(jw)| ||R||. = sup means that -

A:

Q: (a) If X,Y are Banach spaces on XOY _define |(x, y)|| = || +|>ly. Show that this gives a norm making…

A: If X and Y both are Banach spaces then we need to prove that X⊕Y is Banach space with respect with…

Q: Let R be bounded by the hexagon with vertices at the points (2,0), (3,1), (3,5), (2,6), (1,5),…

A: The vertices of the hexagon are A2,0 , B3,1 , C3,5 , D2,6 , E1,5 , F(1,1) Find center of gravity…

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: If D is the disk given by x2 + y² < 4, then it is true that 16л 4 – x² – y² dydx = 3 True False

A: Integration is done by converting function into spherical coordinate system.

Q: Sketch the graph of the convex hull of S. In R, let S =. :0sx<3

A:

Q: Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an…

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Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

A:

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: If x is a normed linear space and the closed Unit ball M is compact, then X is finite dimensional.

A:

Q: Prove that any smooth manifold (M, A) has a countable smooth atlas V (so that D(V) = A)

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.