Show that close ball Y in a metfic space (X,d) is a closed set also show that if (X,d) is complete than (Y,d) is complete

Q: Sketch the set S = {z:|2iz – 2 – 5i| > 2} and state clearly if it is open/closed/neither,…

A: Solution: The set is S=z:2iz-2-5i≥2

Q: let x be a discrete topology Find ent (A) blit int (A) jet AC X and

A: Given that A⊂X where X is a discrete topology. To find: (i) ext (A).             (ii) b(A).…

Q: Sketch a model that satisfies atleast 2 axioms of Fano's Geometry and Identify those axioms.

A: In this type of question, first we need to know, about Axioms of  Fano's Geometry, after this, we…

Q: A(n)................................... is the collection of all points in the plane such that the…

A: We know, A parabola is a collection of all the points in a plane such that the distance from each…

Q: show that the discrete metric on u Vector space X+3 of eannot be obtained from a norm.

A: To Show: Show that the discrete metric on a vector space X ≠ 0 cannot be obtained from a norm.

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

A:

Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

A:

Q: A ° = A – JA

A:

Q: 3) I Consider the topology = {GCR:0 ¢ Gor G = R} on R. Is open %3D %3D interval (-1, 1) compact in…

A:

Q: ) Find the fomily of the equ ation of all lines in 3-space perpen diuler to 2x -y +2u u Given line

A: Family of lines contains parameters.

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: Let (R, d) be the metric space with d(x,y) = |x – yl. Then the open ball B(1,2) is None of these O…

A:

Q: 4).ェ Pare subset, -f y such that CeD than fic) is not eD, then the Convers %3D そイue.

A: Solution As it's mention C=D and f^-1 same , so each and every element from  f inverse we get equal.…

Q: Be M, N subsets of space W. Then span (MUN) = span (M) +span (N)- %3D

A:

Q: If (X.r) and (X,r,) are T,-spaces, then (X.r,nt) is also a T, -space. GOOD LUCK

A:

Q: let (x, m) be a measurable space. then ¥8:% [, o] are measural ble funetion, then ) I +9 is mea…

A:

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

A:

Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: give example is not To-SPace about to folegical SPace

A: We have to find a topological space which is not t0.

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: If S, S, are two subsets of an inner product space V(F) then show that

A:

Q: 3) Consider the topology ↑ {GCR: 0 ¢ G or G = R} on R. Is open = %3D interval (-1, 1) compact in (R,…

A:

Q: Show that R is a bianch space.

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

A: To check whether there exists open ball B(a, r) such that B(a, r) ∩ G= ∅, then whether a is not…

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: b)use Cauchy Schwarz inequality to show that if Let u = and v = inequality to show that a² + b?

A:

Q: Let X be the subset (R x {0}) U (R x {0}) CR2 Show that the quotient space X/~ is locally Eue line…

A:

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: e have to prove that constructed geometric object indeed satisfies three condit

A: To show that given a projective plane P, an affine plane constructed from P denoted as Aff(P) is…

Q: 1) Prove that for any space, X, with any topology, T, that: (i) CI(X) = X and (ii) Int(X) = X.

A:

Q: Let (X,d) be a metric space with the added condition that for any three points x,y,z in X, d(x,y) 0…

A:

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

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

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: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

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: Consider ỹ,,y2, ÿ,ER³ and that y,+y,+y;=0. For Î=span{ ỹ,,y} and Y=span{ỹ2, y, show that Î=Ỹ.

A: Given  that  y→1, y→2, y→3∈ℝ5 and that                   y→1+ y→2+ y→3=0→. and           Y^=spany→1,…

Q: Show that closed ball in metric space (x,d) is a closed set. Also,

A: Given: (x,d) is a metric space The objective is to show that the closed ball y in (x,d) is a closed…

Q: Prove that R" is connected; Prove that [0, 1] × [0, 1] (with the relative topology from R²) is not…

A:

Q: /show that fo-1x is-Contindals everywhere.

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: Consider the topology 7 = {GCR:0 ¢G or G = R} on R. Is open interval (-1, 1) compact in (R, T)?…

A:

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

A:

Q: Give example on the inner product and same time is normed space

A: To find an example on the inner product and normed space. Definitions: Inner product space: Let V be…

Q: Let (X, τ ) be a discrete space. Is (X, τ ) second countable? Explain why

A: Consider the provided question, Let x,τ be a discrete space,    ⇒x,τ  be denote topology if x,τ is…

Q: Exhibit an open cover of the interval (1, 2] that has no finite subcover.

A: An open cover of A is a collection of open subsets of A whose union is all of A and that a finite…

Q: Show that both ℝ and ℝ2 may be covered by countably many open balls

A:

Q: Construct an open cover for the interval [0, 1]. Verify that the open cover is in fact an open…

A: