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Given : X be an infinite set and T be a topology on X such that all infinite subsets of X are open.…

Answered: Let X be an infinite set T be topology…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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establish a bijection between the power set of a set S and the set of functions S—>{0, 1}
Let's say that a collection of open balls (Ui)ie1 = (Bx, (ri))iel in Euclidean space Rd is = UB₂, (ri) iEI and the positive number c>0. If m(U)>c, given, and consider the open set ™ (UU) > m prove that U₁₁, U₁2,..., Uin 3d is established by the presence of each other's element
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Prove that noword G with sounce vertex and Sink ventex t. S be Subset of V seS and teS5=v\s ?
2) Let n E N. Suppose An = {n, n+ 1, n +2, ...} and T = {0, An}neN• (a) Show that t is a topology on N, (b) What are closed sets in t? (c) List the open sets containing the positive integer 6. (d) What is the closure of {4} and {2,4,6, 8, . ..}?
QII State whether the following statements are true or false: 1- Let S be anon-empty subset of the set of real numbers R. ifS is bounded above, then sups is exist but need not to be unique in general. 2- ifA =(-5,5) and B - (5,10), ihen inf (A + B) = 10 and sup(A + B) = 15., 3- the closed interval (1,2] has no maximal element. 4. the set of natural numbers N of R is unbounded. . 3- the set of real numbers R has Sup = o und inf 6- the set 5 = (x €E R;x -25 s 01 has max(S) - 5 and inf(S) = -5 with no minimal element.: 7- the set S= (1+: nez*} has max(5) = 2 and Min(5) = 1. 8- every bounded set of real numbers R has maximal and minimal elements. 9- the properties (M,) and (M2) of the definition of the metric space are state that the distance from any point to another is never negative, and that the distance from appoint to itself is zero. - there are many metric functions d: M x M - R that can be defined on a non-empty set M. -o, 10-
Let τ and τ' be two topologies of a set X. Is the family τ∪τ' formed by the open elements of both topologies also a topology of X? justify your answer
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X, Y and Z be subsets of the set . Verify the following properties: XU(YUZ) = (XUY)UZ, Xn(Y_Z) = (XnY)nz (Associativi XU(YNZ) = (XUY)N(XUZ), XN(Yuz)=(XNY)u(XNZ) XcY+XUY=Y+X Y=X
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Let X be an infinite set T be topology of the a Linfinite subsets that is on x- and let on X in which all Show X are open. discrete topology t