4. Let X= {a,b,c}with topology t, = {ø, X,{a},{a,b}} and Y= {1,2,3} withtopology r, = {6,Y,{1,2},{2},{2,3}}. Define f (a)=1, f(b)=1 and f(c)=2.Verify whether f is an open map.%3D

Answered: Image /qna-images/answer/f96eb22e-d02f-40f4-a73f-a3b908f8e708.jpg

Answered: 4. Let X= {a,b,c}with topology t, = {ø,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 26E: 26. Let and. Prove that for any subset of T of .

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26. Let and. Prove that for any subset of T of .
Let T={X,4,{a},{a, b), {a, c, d}, {a, b, c, d}, {a, b, e}} be a topology on X = {a, b, c, d, e}. (ii) Find N,(neighborhood system of the point c)
Let X={a,b,c} and Y={1,2,3} and t={Y,0,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology L induced by t and f is given by A) L={X,0,{a,c}} B) L={X,0,{a,b}} C) L={X,0{a}} D) L={X,¤} A В C
(2.2) Let C = {a, b, c, d} and T = {0, {a, b, c} , {b, c} , {a},C} be the topology defined on C. (a) Show that the only open sets that are both closed and open in T are C and 0
Let X = {1, 2, 3, 4, 5, 6} and let T1 = {X, 0, {3}, {1,2,3}, {2,3,4}, {2}} and T2 = {X, p, {2}, {1,2,3}, %3D {4,5,6}, {2,4,5,6}} * T1 is not a topology and T2 is a topology on T1 is a topology on X and T2 is not a topology on X O T1 and T2 are both topologies on X T1 and T2 are both not topologies on X
Is T ={∅, X, {1, a}, {1, b}, {2, c}, {1, a, b}, {1}}a topology on X = {1, 2, a, b, c}?Why?
(a) Let M R. Give the radius r and the center c of B(-2,5) n B(6, 7). (b) Let M = R. Using interval notations, give a simplified expression for the set L= [B(1,1) UB(5,1)] n B(3,2). (c) Let = 3(-1)" + and A = {,: ne N). Give the set W of accumulation points of A. (d) Let A₁-(-2n, 2n) and K = U An. Give a simplified expression for the interior Kº of K. NEN (e) Give the set T of isolated points of B(0, 1) U (3) U (2,5,7) (f) Let [P,Q] be a segment in R2 with midpoint H, and let (1,0) and (2, 1) be the components of the points P and H, respectively. Give the components (x, y) of the point Q. (g) Suppose that f: (M, d) →R satisfies d(f(x), f(y)) - vl. Let 21 e M and set n+1 = f(n), for n e N". Assume that M is complete and let a = lim zn. Give a simple formula satisfied by a.
3,4) and TY = Y defined by f(a)=4, f(b) defined by g(a) = 1, g(b)=g(c)=3 and g(d) = 4. Then * Let X= (a, b, C, d} and TX = (Ø, X, {b}, (b,c}, {a,b,d}} be a topology on X. Let Y= {1, 2. (Ø, Y, (3), (1,2,3}} be a topology on Y. Let f be a mapping from X into = 4. Let g be mapping from X into Y = 3, f(c) = 2 and f(d) %3D f and g are not continuous O fis continuous and g is not continuous f is not continuous and g is continuous O fand g are both continuous A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R is equipped with the usual topology, then the boundedness and the closedness are not topological properties because* O la.b] is not homeomorphic to Ja.bl O Ris homeomorphic to1, O[ earch TOSHIBA
4. Let A = {a, b, c}, B = {1,2, 3}, C = {x,Y, z, w}, D = {4, 5, 6}. Let the functions f : A → B, g : B → C, h : C → D be defined by: f = {(a, 2), (b, 1), (c, 2)} g = {(1, y), (2, x), (3, w)} h = {(x, 4), (y, 6), (2, 4), (w, 5)}. Determine if each function is: (а) Onto (b) One-to-One (c) Invertible
Let X = {1, 2, 3, 4, 5, 6} and let T1 = {X, þ, {3}, {1,2,3}, {2,3,4}, {2}} and T2 = {X, þ, {2}, {1,2,3}, {4,5,6}, {2,4,5,6}} * T1 is not a topology and T2 is a topology on O T1 and T2 are both topologies on X O T1 and T2 are both not topologies on X T1 is a topology on X and T2 is not a topology on X

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4. Let X= {a,b,c}with topology t, = {ø, X,{a},{a,b}} and Y= {1,2,3} with topology r, = {6,Y,{1,2},{2},{2,3}}. Define f (a)=1, f(b)=1 and f(c)=2. Verify whether f is an open map. %3D