Remark 2 Suppose f XTx y Tr is a mapping of a topological sp

Remark 2 Suppose f (X,Tx) (y, Tr) is a mapping of a topological space (x,Tx) into a f (Tr)- system of all sets f 1(U) UE Thanks to the point of Lemma 1, we obtain that f-1(TY) is a topology in X. Problem 4 Prove the following theorem: Theorem Let f (x, Tr)-r (Y, a of a space a space (Y, TY). The mapping f is continuous if and only the topology stronger than the topology f \'(Tr). Thanks to point 2 of Lemma 1, we obtain the \"dual\" form of Theorem 3

Solution

X,Tx -----> Y, Ty

X,Tx will be a space into Y,Ty because they are related with an association