Example 2.17. On N, let {{0,1}, {2,3}, {4, 5},..} be a subset of a topology Tad. What other sets must be in Tad at a minimum for it to be a topology on N?

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), n=(1,2,3.4), Ty:= (0,Y, {1,3), [3,4}, {3), [1,3,4}}. Let fx-Y. f:= {(a,3). (b, 3).…

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Q: 3.1 Show that an infinite intersection of closed sets E k = 1,2,3,., in a metric space (SA) is a…

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Q: Let T = {0, En}, where En = {n +1, n + 2, n + 3, ...} with n E N, be a topology on N. Then the set…

A: Ans : False , because

Q: 2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite…

A: It is given in part (a) that (X,t) be any topological space, and the interval (x-A,x+A) is only open…

Q: Q3) Discuss these examples (a) Let X be an infinite set and t is the cofinite topology on X. Then…

A: We shall solve these problems in next step. Please read the answer carefully and slowly to…

Q: 1) Let X= {a, b, c} List the open sets in the following topologies on X: (i) The discrete topology;…

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Q: 15. If t={X,Ø,{1,4},{4,3},{1,4,a},{b}} is a topology on X-{1,2,3,4}, then: * a=4, b=2 O a=3, b=1 O…

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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: I? B is a family of subsets of X Such that: (2) If B, ,BzE/ß them B,nB, EB Then B is a base for a…

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Q: 7. Let X = {a,b,c}. Show that Bß ={{a,b},{b,c}}cannot be a base for any topology on Х.

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Q: Example 2.3 Let A be a set with A = n <x. How many distinct subsets does A have?

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Q: If A = {a, b, c, d) and B = {c, d, e, f, g}, then n(A U B) = n({a, b, c, d, e, f, g}) = 7, n(A) = 4,…

A: We have to find

Q: Q2: Let the set under consideration be N for each n EN, define u, = {n, n+1, n+2, ...} and let 1 =…

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Q: 4. Let X be any set and 7 be the countable complement topology on X. b) If X is uncountable, then…

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Q: 5. If {T,} is a family of topologies on X, show that Ta is a topology on X. Is UT, a topology on X?…

A: First we will prove the result that arbitrary intersection of a topologies is a topology by showing…

Q: Theorem 3.16. Suppose X is a set and S is a collection of subsets of X. Then S is a subbasis for…

A: Given that X is a set and S is a collection of subsets of X. Let B be the family of finite…

Q: 1.2.6. Show that the following sequences of sets, {Ck}, are nondecreasing, (1.2.16), chen find…

A: Since you have asked multiple question we will solve the first question for you. If you want any…

Q: 1, 2, 3, 4 LeT X = find The Possible Topology That Contain seas { 1123 , 2, 3, 1, 3, 4

A: Given : X={1, 2, 3, 4} To Find :  possible topology that contain {1 ,2 } ,{ 2, 3} ,{1 ,3 , 4} .

Q: 3 let X= {ab,c},B =} {n,b}, {9c3}.5how that Whether B is a %3D %3D base for any topology on X or not

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Q: 1. Determine if the given collection of subset of X={a, b, c} is a topology on X. a. T1 = {Ø, X,…

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Q: (4.2) Let X = {1,2, 3, 4} and 7 = {0, {1,2,3} , {2, 3},{1},X} be the topology defined on X. (a) Show…

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Q: Let n, k E Nand define the family of graphs Gnk= (Vn, Enk) on the vertex set %3D Vn = {1,2, ...,n}…

A: Given family of graphs is Gn,k=Vn, En,k on vertex set Vn=1, 2, 3,..., n with edge set is…

Q: 3) (i) Let X be a set containing elements a and b. Define T= {X, Ø, U| a E U and b E U}. This is a…

A: Definition:  A collection of subsets I of X is said to be a topology if it satisfies following…

Q: 2. Draw a Hasse diagram for the POSET(S,R) where S = {1,2,6,8,12,18,48,72,108,144,216,432} and R be…

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Q: 4.4. Let S be a nonempty set. Suppose that to each ordered pair (x, y) E S x S a nonnegative real…

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Q: 2. Recall that for two sets A, B, the Cartesian product of A and B is the set A × B = {(a, b) : a E…

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Q: d) The set E = (-1,0) U (0, 1] is an open set. %3D

A: Given set is E = (-1, 0) ∪ (0, 1]

Q: 4) Describe the graph of the given set C o{ (x,y): ニ

A: Parametric curves: When we define the coordinates of the point x, y in terms of another variable,…

Q: Question 1. Let X and Y be infinite sets. Let X be equipped with the discrete topology, and Y…

A: Given, X and Y be two infinite sets in topological space.  X is a discrete topology and Y is a…

Q: 3.1.10. Let M and N be matchings in a graph G, with |M| > |N|. Prove that there exist matchings M'…

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Q: Example 2.30. Let T be the topology on N which contains Ø, N, and for each N E N with N > 0, the set…

A: Definition: Let X be a non empty set and τ be a collection of subsets of X. Then X, τ is said to be…

Q: 1.39. For each point (m, n) in the digital plane, determine the smallest closed set containing (m,…

A: given:  for an each point  (m, n) in the digital planeTo…

Q: 4) Determine whether or not each of the following subset of R with the euclidean topology is…

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Q: 10. If t={X,Ø,{2,4),{1,4},(1,2,4},{4}} is a topology on X={1,2,3,4}, let A={2}, then the closure of…

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Q: (2.1) Let b> 0 and X [-b, b). Here we admit %3D B"open" in r if and only if (0} B or (-b, b) C B.…

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Q: If A= {a, b, c, d} and B= {c, d, e, f, g}, then n(AU B) = n({a, b, c, d, e, f, g}) = 7, n(A) = 4,…

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Q: For A = {a, b, c}, let (24, S) be a partially ordered set. Find (i) an upper bound, (ii) a lower…

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Q: 1. Consider the relation < on the set {1,2, 3, 4}. Write down this relation as a set of ordered…

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Q: 7. Suppose that a set X is equipped with the finite closed topology. Which one of the followings is…

A: As per the policy, we are solving first question. Please repost the question and specify which…

Q: Q. 1: If X = (a, b, c, d, e) Determine whether or not each of the following collections of subsets…

A: Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X: 1) Both the…

Q: If A = (0, 1) n Q, then {x; x E (0, 1)} is the set of all cluster points of A. Select one: OTrue…

A: Given A= (0,1) ∩ Q . Then the set of all cluster points of A is  = {x: x ∈ [0,1]}

Q: Let X = {1, 2, 3, 4, 5, 6} and let T1 = {X, p, {3}, {1,2,3}, {2,3,4}, {2}} and T2 = {X, ¢, {2},…

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Q: Suppose {p, p,, P3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show…

A: Here, {p1,p2,p3} is an affinely independent set. So, {p1,p2,p3} are linearly independent.

Q: ) We know that {-5,-4,-3,-2,-1} and N are wellordered sets with respect to the trivial ordering. Let…

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Q: 2. Find the minimal and maximal element of the given Hasse diagram. Consider the subsets {b,c} {g,…

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Q: Question 3 Show that a subset G of R" is open iff it is the union of a countable collection of open…

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Q: Let T = {X. Ø, {b}, {a, b}} be a topology on X = {a,b,c} and let A = {a,b,c}, B = {a,b,c}. Find a)…

A: Given That: Let T={X,ϕ,{b},{a,b}} be a Topology on X={a,b,c} A={a,b,c} ,B={a,b,c} To Find: a)…

Q: 3.1 Show that an infinite intersection of closed sets F, k 1,2, 3,..., in a metric snace (S, d) is a…

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Q: Let X = {a, b, c}, T1 = {X, 0, {a}, {b}, {a,b}} and T2 = {X, 6, {a}, {c}, {a,c}}. Then one of the…

A: According to experts guidelines of bartleby i have to solve only first problem so repost for further…

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.