e Guen an example to shus thet the inege lincarly independant set under a Lucar may mapping not be lineaily independat.

Q: If  X is a countable set show that every topology on X is second countable.

A: To Show: If  X is a countable set show that every topology on X is second countable.

Q: Let X = {a, b, c, d} with the following topology: ?= {X,0, {a},{b}, {a,b}} Let Y={x,y,z} Consider…

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Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

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Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

Q: Queston2 {e-x, xe-O onlo, c∞) a) shau that q fandamented Set

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A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

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A: A relation R on a set X is said to be anti-reflective if no element  of X is related  to itself…

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the finite…

A: Answer

Q: Determine whether is a linearly independent set of vec- 3 11 tors in R3.

A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…

Q: Let X:= {a, b, c}, Y:= {1,2,3,4}, Ty: = {0, Y,{1,3}, {3,4}, {3}, {1,3,4}}. Let f:X → Y, f:= {(a, 3),…

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Q: We defined the relation between sets by A ~ B means that there there is a 1-1 correspondence ƒ : A →…

A: We define the relation ~ between sets by A~B means that there is a one-one correspondence f:A→B Show…

Q: Consider RxR to be a universal set with subsets A, B, and C defined as follows. A = {(x,y)\x² +y°…

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Q: let Then Show thato- Prelurt Space IR? an Inneo on

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Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

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Q: 5. If r = (r, (x, y, z) and ro = (xo, Yo, 20), describe precisely the set of all points (r, y, z)…

A: We plug the given values in the given equation and simplify

Q: If A C BỊ U B2 where B1 and B2 are disjoint open sets and A is compact, show that An B1 is compact.…

A: To Show- If A ⊆ B1 ∪ B2 where B1 and B2 are disjoint open sets and A is compact, show that A ∩ B1 is…

Q: A non-empty set A in the standard topology on R so that ðA= 0.

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Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: Which one of the following statements is true? O R with the Euclidean topology and R with the…

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Q: given two sets X={-1,0,1} and Y={u,v,x,y,z}. There are 3x(3-1)x(3-2)%36 onto mappings from X to Y.…

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Q: Let R be the relation on set {a,b,c,d} containing ordered pairs (a,c),(b,b),(b,a)(c,a)(c,d) and…

A: Here given that the set is A = {a, b, c, d} and the relation on A is given by R = {(a, c), (b, b),…

Q: Ler be a topology on X. Assume that is Husdouff and let xe X. 1. Show that {x} (and hence every…

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Q: Find a topology in which the point x = 5 is an open set.

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Q: A relation R defined on the given set A will satisfy

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Q: Which one of the following statements is true? * R with the Euclidean topology and R with the…

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Q: Let ā = (- 4, - 5) and b (- 5, 0). Find the projection of b onto ä. Round your answer to at least 3…

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Q: . Show that every map f : A → B of sets decomposes as hog where g : A → C is surjective and h : C →…

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Q: - Is T a one to one mapping?

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Q: The closare of a set is smallest closed set containing s صواب ihi

A: Let S be a subset of X: S⊆X. Let M be the closure of S. Then M is the smallest closed set which…

Q: If X = {u,v} what is the power set of X? (in answering this question, use {} as the null set, and do…

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Q: Let R be a relation defined on the set Z by aRb if floor value of a = floor value of b. What…

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Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: Show that the interval [0,1] is a closed set in R

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Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then…

A: X is finite

Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

Q: If fis a bijection, find the sets X and Y.

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Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: EX LeT X = {!,2,3,415,6} T- Ex,¢,てり,E4,しろ4), [ス456 Topology.?

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Q: (c) A line in R² not through the origin ax then ŏ ¢ H. : Itis not asubspace Umecepsung but…

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Q: Let R be a relation from set A to B, if R is symmetric then its inverse will be...... Answer (Please…

A: Given R is a relation from set A to B. Let R is symmetric relation and R-1 be the inverse relation…

Q: Determine if the set S is span set of %3D R2x2. Solution:

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Q: (2.2) Let C = {a, b, c, d} and T = {0, {a, b, c} , {b, c} , {a},C} be the topology defined on C. %3D…

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Q: A relationR is called a partially ordered set if R is: .a reflexive .b antisymmetric .C transitive…

A:

Q: Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)}…

A: Given that A is non empty set which is bounded below by 3. Given that B=x∈R|∃a∈A,b=1a The objective…

Q: We defined the relation between sets by A - B means that there there is a 1-1 correspondence f : A →…

A: The complete solutions is in given below