Let T us be the usual topology on R and Tl be the lower limit topology generated by the unions of (Ja,b]/ a,beRasb}. Define f the mapping from (R. T us) into (R, TI) by f(x) - 1 and g the mapping from (R. T ) Into (R, T_us) by g(x) = 4x. Then* O gis a homeomorphism butf is not O fis a homeomorphism but gis not f and g are both homeomorphisms O"one of the choices

Let T us be the usual topology on R and Tl be the lower limit topology generated by the unions of (Ja,b]/ a,beRasb}. Define f the mapping from (R. T us) into (R, TI) by f(x) - 1 and g the mapping from (R. T ) Into (R, T_us) by g(x) = 4x. Then* O gis a homeomorphism butf is not O fis a homeomorphism but gis not f and g are both homeomorphisms O"one of the choices

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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Q2: a) Let f be a map from a set X into a space (Y,T) and T' {Us XaV e T,U f"(V).show that T is smallest topology on X,makes f continuous. b) Let T be a co finite topology on a set of reai numbers R and f: R →R be a map such that f(x) =k + 2/, show that whether f is T-T continuous or not.
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Let T_us be the usual topology on R and TIl be the lower limit topology generated by the unions of {Ja,b]/ a,beR;asb}. Define f the mapping from (R, T_us) into (R, T_II) by f(x) = 2x and g the mapping from (R, T_II) into (R, T_us) by g(x) = 3x. Then * %3D O f and g are both homeomorphisms f is a homeomorphism but g is not g is a homeomorphism but f is not O None of the choices
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