sols25

sols25 - 58 25. December 6th: Topology III. 25.1. Let (T1 ,...

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58 25. December 6th: Topology III. 25.1 . Let ( T 1 , U 1 ), ( T 2 , U 2 ), and ( T 3 , U 3 )betopo log ica lspaces ,andlet f : T 1 T 2 , g : T 2 T 3 be functions. Prove that if f and g are both continuous, then g f is continuous. Answer: By DeFnition 24.1, in order to verify that g f is continuous we must verify that ( g f ) 1 ( U )i sopenin T 1 for every open set U of T 3 . Now, since g is continuous, then for every open set U of T 3 we have that g 1 ( U ) is open in T 2 .I t follows that since f is continuous, then f 1 ( g 1 ( U )) is open in T 1 . Now ( g f ) 1 ( U )= f 1 ( g 1 ( U )) , so indeed ( g f ) 1 ( U )isopenin T 1 for every open set U of T 3 ,asneeded . ° Note the funny reversal in the order of f and g in the last display. If this seems confusing, verify the equality formally: x ( g f ) 1 ( U )meansthat( g f )( x ) U , that is, g ( f ( x )) U ,thatis , f ( x ) g 1 ( U ), etc. 25.2 . Let ( T, U )beatopo log ica lspace ,andlet S
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sols25 - 58 25. December 6th: Topology III. 25.1. Let (T1 ,...

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