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State and Prove closed Graph Theorem.

Answer»

Inmathematics, theclosed graph theoremis a basic result which characterizescontinuous functionsin terms of theirgraphs. There are several versions of the theorem.

The closed graph theoremIn point-set topology

For any functionT:X→Y, we define thegraphofTto be the set

Inpoint-set topology, the closed graph theorem states the following: IfXis atopological spaceandYis acompactHausdorff space, then the graph ofTis closed if and only ifTiscontinuous.

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