Let f : (X,Tx)→ (Y,TY) be afunction,where TX is a topology on Xand TYis a topology on Y.Show that if the inverse image ofa)every basic open set is open then thefunction f is continuous.b) IfX =Y = R, Ty ={VCY;f(V) E TX}%3D-3,а <0and f(x) = { 3,0 < <9 .%3D6,and the topology on X is the Left raytopology.

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 27E: 27. Let , where and are nonempty. Prove that has the property that for every subset of if and...

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Let f : (X,Tx)→ (Y,TY) be a function,where TX is a topology on X and TY is a topology on Y. a) Show that if the inverse image of every basic open set is open then the function f is continuous.