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Posted: **Thu May 05, 2022 6:33 pm**

: Let X T Y S And F Be As In Problem 1 Show That T Is The Coarsest Smallest Such Topology In Which F Is Continu 1 (24.28 KiB) Viewed 45 times

: Let X T Y S And F Be As In Problem 1 Show That T Is The Coarsest Smallest Such Topology In Which F Is Continu 2 (41.27 KiB) Viewed 45 times

Let (X, T), (Y,S), and ƒ be as in Problem #1. Show that T is the coarsest (smallest) such topology in which f is continuous with respect to T and S.
Let X be any set and (Y, S) be a topological space (where Y is a set and S≤P(Y) is a topology on Y). Let f:X→Y be any function and let us define T := {f-¹(V) ≤X|V€S} ≤P(X). Show that T is a topology on X and f is continuous with respect to T and S.