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# sols25 - 58 25 December 6th Topology III 25.1 Let(T1 U1(T2...

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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 ) be topological spaces, and let 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 Definition 24.1, in order to verify that g f is continuous we must verify that ( g f ) 1 ( U ) is open in 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 . It 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 ) is open in T 1 for every open set U of T 3 , as needed. 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 ) means that ( g f )( x ) U , that is, g ( f ( x )) U , that is, f ( x ) g 1 ( U ), etc. 25.2 . Let ( T, U ) be a topological space, and let S T be a subset. Verify that the collection of sets U S , as U U , forms a topology.

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