1- Suppose that ,p) is a meric space, an that E CX. Then the interior E is open (m X) and the clesure E is closed (in X).

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: det (NT) is sPace Toflogecu! and T=0U{un}such that 7Show that (N, Cl is compact slase

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: Suppose M and N are C manifolds, U an is C. Show that there exists a neighborhoo that F can be…

A: Let us start solving the problem step by step

Q: Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded

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Q: (a) Suppose that (M, d) is a metric space. Consider the mapping f: M [0, 1] be given and let di(x,…

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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…

A: TRUE

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: Let x be O and all subsels f X whose complements Prove that (X, z) infimite set and let z consist of…

A: Consider, I=A⊂X : Ac is countable ∪ϕSince,Xc=ϕ is countable, X∈I Also given, ϕ∈I Assume, Aii∈N∈I…

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 (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: 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: Q2: In usual topological space (R.T), answer the following: 4- Is [0,1] is arcwise in R? Why?

A: Given that ℝ, Tu is usual topological space. To show: 0,1 is arcwise in ℝ.

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: E be a Euclidean space of dimension (E(1, f)) = dim(Ker (f- id)), then s minimal.

A:

Q: Suppose Show that p: X²> t0,00) defined by d is a metric n X, 2. 3d (x,y) 2+3d (xy) metric an p…

A: Given that d is a metric on X. We need to prove that ρ is a metric.

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

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Q: 4. Consider the space (R,T), where T={R,Ø}u{AcR:[0,1)cA}. If A=(0,1), then the closure of A is: * O…

A:

Q: Suppose (X, d) is a metric space and let r> 0. Show that p: X2 → [0, c0) defined by p(x, y) = r·…

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

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Q: 4. Consider the space (R,T), where T={R,Ø}u{AcR:[0,1)cA}. If A=(0,1), then the closure of A is:…

A: consider the space (IR,T)where, T = IR,ϕ∪A⊂IR[0,1]⊂AIf A = (0,1), then closure of A is

Q: Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M);…

A: Given as X,T be a topological space,M,d be a complete metric space and BCX,M=f∈CX,M;fX is bounded…

Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

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Q: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

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Q: 3.4 Give the definition of a separable metric space (S, d). 3.5 Show that the metric space (R, d),…

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Q: Let (X, d) be a metric space and B = S(xo,r) be the open ball with centre at xo E X and radius r> 0.…

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

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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: Let f(x,y,z) = zIn(x³ycos(z)), find Z 3x² O In(x3ycos(z)) - ztan(z) 0 2 Y о 3z X af ila

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Q: 2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1,…

A: To prove that every Cauchy sequence in the product space XxY converges (under the metric defined in…

Q: 4. Consider the space (IR,T), where T={R,Ø}u{AcR:[0,1)cA}. If A=(0,1), then the closure of A is:…

A:

Q: a. Show that T: 1² → 1² defined by Tx = y = (n), n, = r, is compact where x = (5,) and where0 <r< 1.

A: Show that T: l2→l2 defined by Tx=y=ηj, ηj=rjξj is compact where x=ξj and 0&lt;r&lt;1. Definition:…

Q: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

A: Given

Q: Consider a Z endowed with the metric if r = y d(r, y): 1979-" if r # y Where n is the largest…

A: According to the given information consider Z endowed with metric space:

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

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Q: Let X,d) be a metrie space. Define d': Xx X >R by: d'Cxoy) Show that (X.d') is min (dl8), 1). a…

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Q: Please solve this linear algebra question. Thank you.

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Q: Let (X, || · ||) be a Banach space, J CR be bounded and F C C1 (J, X ). If sup ||f(x)|| < , f€F, rɛJ…

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

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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: Suppose the matri x A is 27 X 36 and the rank of A is 14. Find the nullity of A a e dimension of…

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Q: 3 \et F be anormid space and- |2+j|l <llxll+yllprove that F has a stridtl (3) y convex- orm

A: We know that a normed space is strictly convex if and only if x≠y, ∥x∥=∥y∥=1, we have ∥x+y∥&lt;2 Let…

Q: O let d:Rx R R be defned byd(x)= (x-, Ha,ysR (Rd) is a wmetric space. Rd) Ihen

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Q: if x is a compact Metric space and F e C (X₂Y) is equicontinuous, then F i's uniformly…

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Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…

Q: 2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined by ρ(x, y) = d(x, y) 1 +…

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

A: