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I need some help in order to fully understand Kalajdzievski's definition of a closed set in a topological space ...

The relevant text reads as follows:

As I understand it many closed subsets of the underlying set \(\displaystyle X\) of a topological space \(\displaystyle (X, \tau)\) do not belong to the topological space because they are not open ... i.e. they are not clopen sets ...

Is my interpretation of the above situation correct ... ... ?

Help will be appreciated ...

Peter

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It may help readers of the above post to have available Kalajdzievski's definition of a topological space ... so I am providing the same ... as follows: