2. Let A, B be non-empty sets in a metric space (X, d). Define p(A, B)=inf{(a,b): a e A, b e B}.i.non-emptysubset(X, d).provethatIf SS = {x:x € X and p({x}, S) = 0.If S is any non-empty subset of (X, d), prove that the function f: (X,d) → Rdefined by f(x) = p({x}, S), x e X is continuous.S isanyi.%3Dof 1. Determine if the given collection of subset of X={a, b, c} is a topology on X.a. T1 = {Ø, X, {a}, {b}}.b. T2 = {Ø, X, {a}, {a, b}}c. T3 = {Ø, X, (b), {b, c}}.%D

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Answered: 1. Determine if the given collection of…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 22E

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1. Determine if the given collection of subset of X={a, b, c} is a topology on X. a. T1 = {Ø, X, {a}, {b}}. b. T2 = {Ø, X, {a}, (a, b}} c. T3 = (Ø, X, (b), {b, c}}.