Let B {(x,y, z) E R: -1< y +x² < 1, 0 < z – r' < 2} CR. %3D (a) Prove that B is a closed subset of R3. Hint: use the fact that the inverse image of a closed set under a continuous mapping is closed. (b) Is B compact? Why or why not?

Q: Prove that If f is holomorphic on D(0, 2), f(0) = -i and |ƒ(z)| > 1 on the unit circle then f has at…

A: It is given that f is holomorphic in the disc with centre 0 and radius 2. So, the function f is…

Q: if and only if F-1: R(F) →X is bounded.

A:

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

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

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

Q: How do you know that for all a, b, c, d: ax + bx will be orthogonal to cx? + d ?

A:

Q: Let A and B be topological spaces and endow A x B and B x A with respective product topologies. Show…

A: Homeomorphism: Let X and Y be topological spaces; let f:X→Y be a bijection. If both f and f-1:Y→X…

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. * None…

A:

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: 1.4.10. Prove that the closed interval [0, 1] and the open interval (0,1) have the same cardinality…

A: (Schr¨oder-Bernstein theorem) :Let A and B be sets. Suppose there are injective functions f : A → B…

Q: 5. Consider the function f (X, J) → (Y, U), where (X, T) and (Y, U) are two topological spaces.…

A:

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

A:

Q: Q3 Assume that (M,d) is a Compact metrie Space. Show that : (M,d) → (ya) is Continous bjective then…

A:

Q: Let (X, Tx) represent a topological space and {a} a singleton set, which is an open subset of X, {a}…

A:

Q: 3. Let f: X→ Y be a surjective, continuous map of topological spaces. Prove that if X is compact and…

A:

Q: Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a…

A: Let f:X,Tx→Y,Ty be a function and a is open in Tx. To prove that the function is continues at a.…

Q: A) Let (R, T) be the usual topological space and f: R → R s.t. f(x) = 2x + 2.Show that f is…

A: Given that (ℝ,τ) is the usual topological space (the standard euclidean topology over ℝ). Also given…

Q: Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if…

A: This is a problem of topology.

Q: Let X and Y be topological spaces and endow X×Y and Y×X with the respective product topologies. Show…

A: To Show : Let X and Y be topological spaces and endow X×Y and Y×X with the respective product…

Q: 0 IS SU b) Let f be a map. from a space X ipto a space Y and 6 is a base for Y .show that f…

A:

Q: (v) Prove that if the space (X,T) has the fixed-point property and (Y, T1) is a space homeomorphic…

A: Given: v) To prove the given statement as follows,

Q: Suppose A is a function that assigns to each subset of the plane a number in such a way that the…

A: Given that AX≥0 and if X and Y are disjoints then AX∪Y=AX+AY To prove if X⊂Y then AX≤AY It is known…

Q: Let B = {(p,q) e R² : p² + q² = 1}. (a) Show that B is NOT disconnected (b) Given that there is no…

A: Given: B=p,q∈ℝ2 : p2+q2=1. There is no continuous mapping f:B→ℝ. To prove: B is not disconnected.…

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (IR, 71) → (R,…

A:

Q: Let X ų be topological spaces and f: X→Y be a continuous map. If X is completely normal then Y is…

A:

Q: Let fbe a continuous function on an open subset U of a Banach spe into a Banach space Y. Let a and b…

A:

Q: (b) Let U, T be usual and cofinite topologies on se:t of real numbers R and f be a map from R into R…

A: As per our guidelines we are supposed to answer only first question. So i have solve  question (b)…

Q: Let (IR, I) be usual 2 2 IR ² X IR² = Cx C space. Prove

A: Solution:  The  given equation f:ℝ2→ℂ   defined  by  f(a,b)=(a+ib), a,b∈ℝ2. Then,a,b∈ℝ2, and…

Q: Show that the map T: R³ R² given by T(X₁, X₂, X3) = (X₁ + X₂, X3) is an oper map.

A:

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 f be a continuous functio to R. Let a and b be two distinct ntained in U and fbe differentiable…

A:

Q: Suppose A is a closed, non-empty subset of a metric space (X, d) and p a point of X which is not in…

A: Given:  A is a closed and non-empty set of a metric space (X,d) and p∈X , p∉A. To Prove: inf{d(p,a)…

Q: Let V = F" where FE {C, R} show that V is a Banach space with the norm ||x| = maxisisn{xil VxEF".

A:

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

A:

Q: If A, B are subsets of the topological space X show that Ext(AUB) = Ext(A)n Ext(B).

A: Given: A and B are subsets of the topological space X. To Prove: ExtA∪B=ExtA∩ExtB

Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

A:

Q: 5)Evaluate F - nds for F = y²zi + y°j+ xzk using Gauss' Theorem. Consider that S is the boundary of…

A: We will find out the required value.

Q: 5.1: The following stat- X is disconnected.

A:

Q: 3. Let X be a topological space and let f: X→ Sn a continuous map to the n-sphere. Prove the…

A: Given That : Let X be a topological space and f:X→Sn a continuous map to the n-sphere . Prove That :…

Q: Prove that (F,|.|) is Banach space where FE{C, R}

A: Definition required:  Let X be a metric space and let xn be a sequence of points in X. (a) Cauchy…

Q: Let f : (X,d) - (Y, d1) becontinuous fumction, then for any closed set G in X, f(G) is not necessary…

A: False