2. Find the complement CC of the set C with respect to the space C if (a) C = {x :0 < x < 1}, C = {x : < x < 1}. (b) C = {(x, y, z) : x²+ y² + z² < 1}, C={(x, y, 2) : x² + y² + z² = 1}.

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

A:

Q: or not. (c) Let f be a map from a space(X, ). into a space (Y,T) and t,T such that TSt and TS,, show…

A:

Q: 6. Show that a compact metric space X is locally compact.

A: The given problem is related with metric space. We have to show that a compact metric space X is…

Q: 10. Let (X, 71) be a Lindelöf space and (Y, T2) be a compact space. Then show that the carte- sian…

A:

Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

Q: If X/M ={ [x], xe Xx} show that X/M is a linear space, Where [x] is equivalence class

A: To show that X/M is a linear space, we need to verify that it satisfies the following axioms:…

Q: Y Show that f (2)= 2+ Con formal maps %3D on %3D

A: Given that,f(z)=z+1zz=1

Q: Suppose that X and Y are topological spaces and f: X- Y. Show that f is continuous iff for each…

A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

Q: 3. Let (X, 7) be a Hausdorff space. If X is finite, then show that 7 = P(X), where P(X) is a power…

A:

Q: Consider the set V = {(x, y) = R² : x+y=1}. on this set defined in the following way: (x2, y2) = (x₁…

A:

Q: Dijec not need to prove you function is a bijection): (a) N and Z (b) Rand (1, ∞) (c) (0, 2) and (3,…

A:

Q: Let X be a topological space

A:

Q: let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B…

A: Let (X,t)  be  not  connected.We  have  to  show  that  X=A∪B ,where  A and B are  elements  in t…

Q: Show the 3-dimensional ball B₁: ((..) ER³ | x² + y² +2² 0.

A:

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 9 an…

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

A:

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

A:

Q: of the quotient space R²/ given below, determine wheth under the quotient projection p: R² → R²/.…

A: (a) The set {[(0, 0)]} is the set of all equivalence classes that contain the point (0,0). The…

Q: Show that the dual space of the space co is 1¹.

A: Statement:Dual space of the space  is .Definition:a) The space  is the set of all sequences  such…

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

A:

Q: 7. Sketch the set {(x, y) = R² : ||(x, y) || = 1} when: (a) ||(x, y)=√√√x² + y² (b) ||(x, y) || =…

A:

Q: 23. Determine whether each of these functions is a bijection from R to R. a) f(x) = 2x + 1 b) f(x) =…

A:

Q: Q2:- Let (X, d) be a metric space. Xo, Yo EX and r, s > 0. Show that B, (yo) Br(xo) if s+d(xo, yo)…

A:

Q: True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all real numbers with the…

A: The given statement is  Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with…

Q: R= {(0,2). (1, 1). (1,3), (2, 1). (2, 2). (2,3 For relation R find the Reflexive closure; Symmetric…

A: Given:- Set A={0,1,2,3}   , relation R:A×A is defined as : R={…

Q: = Prove the operation of addition on the set Q = {[(x, y)] : x ≤ Z, y ≤ N} defined by [(x, y)] +…

A: Let us show that given operation is well defined.

Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

A:

Q: 15. Let x = (x1, x2). Using the feature mapping show that *(2,3)·¤((4,4) –-(2,3)·(4,4)*

A: Given that for x=x1,x2, the feature mapping is…

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

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: 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: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

A:

Q: 11. Prove that the function d: Cx CR defined by d(x, y) = Solle is a metric on the set of all…

A:

Q: 5. In any metric space (M, D), prove that D(a1, an) ≤ D(a₁, a2) + D(a2, a3) + + D(an-1, an), for a1,…

A: Given M,D is a metric space.   The triangle inequality holds in a metric space, which means for…

Q: Let X be a set with more than one element. (X,7) is a topological space such that Vxe X, {x} €T then…

A: Given the set X contains more than one element.

Q: 3. Show that is a closed subset of R². {(x, y) = R²: y = x²}

A:

Q: We note that ℤ2 which means ℤ*ℤ. Let d: ℤ2 *ℤ2→ℝ≥0 be the function that is given by…

A: The given in this question is a function d that takes two points (x0,y0) and (x1,y1) in the set of…

Q: For each of the sets find the set of boundary points and the set of interior points. Also state if…

A: We have, (a) [0,1]U{4} (b) x,y∈ℝ2:x2+y2=1 Here we have to find the  o find: Interior Boundary…