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Chapter 1 / Re: why complex plane closed
« on: October 01, 2020, 09:37:19 PM »
You can also use the different properties to argue that both the empty set and complex plane are open and by complementarity both are closed. Specifically referring to those listed in JB's lecture as attatched,
(1) says that the set S is open iff the intersection between S and its boundary is the empty set. For the empty set itself, the intersection between the empty set and its boundary (also the empty set) is the empty set. So the empty set is open.
By (2), the complex plane is a neighborhood of every element in the complex plane by definition. so (2) is satisfied.
By (4), since the complement of the empty set is the entire plane, then the entire complex plane must also be closed. Vice versa to conclude that both the empty set and complex plane are open and closed at the same time.
(1) says that the set S is open iff the intersection between S and its boundary is the empty set. For the empty set itself, the intersection between the empty set and its boundary (also the empty set) is the empty set. So the empty set is open.
By (2), the complex plane is a neighborhood of every element in the complex plane by definition. so (2) is satisfied.
By (4), since the complement of the empty set is the entire plane, then the entire complex plane must also be closed. Vice versa to conclude that both the empty set and complex plane are open and closed at the same time.