Q1lal Let X be an arbitrary infinite set and let r the family of all subsetsF of X which do not contain a particular point x, EX and thecomplements F of all finite subsets F of X show that (X.r) is a topology.bl The nbhd system N(x) at x in a topological space X has the followingpropertiesNO- N(x) for any xXN1- If N EN(x) then x€NN2- If NEN(x), NCM then MeN(x)N3- If NEN(x), MEN(x) then NOMEN(x)N4- If N = N(x) then 3M = N(x) such that MCN then MeN(y) for anyуемShow that there exist a unique topology τ on X.Q2\a\let (X,r) be the topology space and BST show that ẞ is base for atopology on X iff for any G open set xEG then there exist A Eẞ suchthat x E ACG.b\Let ẞ is a collection of open sets in X show that is base for atopology on X iff for each xex the collection B, (BEB\xEB) is is anbhd base at x.-Q31 Choose only two:al Let A be a subspace of a space X show that FCA is closed iffF KOA, K is closed set in X.الرياضياتb\ Let X and Y be two topological space and f:X - Y show that'f iscontinuous iff for each closed set K in Y f¹(K) closed set in X.14. 05. 2024

Q1(a)Problem:Let X be an arbitrary infinite set, and let τ be the family of all subsets F of…

Answered: Q1lal Let X be an arbitrary infinite…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 20E: Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not...

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Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.
29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there is a subset such that . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets .
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A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
Q1\a\ Let X be an arbitrary infinite set and let the family of all subsets F of X which do not contain a particular point x, EX and the complements FC of all finite subsets F of X show that (X,T) is a topology. b\ The nbhd system N(x) at x in a topological space X has the following properties NO- N(x) for any x Є X N1- If NE N(x) then x E N N2- If NE N(x), NC M then M = N(x) N3- If NE N(x), ME N(x)then NO MEN(x) T=ACX, 4*, AC Smit] N4- If N EN(x) then 3 MEN(x) such that MCN then MEN(y) for any Show that there exist a unique topology τ on X.
Q1\a\ Let X be an arbitrary infinite set and let the family of all subsets F of X which do not contain a particular point x, EX and the complements FC of all finite subsets F of X show that (X,T) is a topology. b\ The nbhd system N(x) at x in a topological space X has the following properties NO- N(x) for any x E X N1- If NE N(x) then x E N N2- If NE N(x), NC M then M = N(x) N3- If NEN(x), M = N(x) then NO MEN(x) T-AEX, 4*, *S] N4- If NEN(x) then 3M EN(x) such that MCN then MEN(y) for any yЄM Show that there exist a unique topology τ on X. Q2\a\let (X,T) be the topology space and ẞET show that ẞ is base for a topology on X iff for any G open set,x E G then there exist A = ẞ such that x E A ≤ G. b\Let ẞ is a collection of open sets in X show that ẞ is base for a topology on X iff for each x EX the collection ẞx = {BEẞ \xЄ B} is is a nbhd base at x. Q31 Choose only two: al Let A be a subspace of a space X show that FSA is closed iff
Consider X = {a1, a2, ..., an}Can a topology of X with 3 open and one with 4 open be equivalent?
Q1 Let N be the set of all positive integers. Prove that each of the following collections of subsets of N is a topology. (i) Tị consists of N, Ø, and every set {1,2, ..., n}, for n any positive integer. (This is called the initial segment topology.) (ii) T2 consists of N, Ø, and every set {n, n+1,. }, for n any positive integer. ... (This is called the final segment topology.)
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