Let X be a set. Consider the following family T of subsets of X: T =...

 

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Let X be a set. Consider the following family T of subsets of X: T = {0}U {OCX | XO is a countable set}. (The family T is called the countable complement topology.) (a) Show that T is a topology. (b) Show that if AC X is a countable subset, then the derived set A' is empty. (c) Show that if AC X is an uncountable subset, then A' = X. Let X be a set. Consider the following family T of subsets of X: T = {0}U {OCX | XO is a countable set}. (The family T is called the countable complement topology.) (a) Show that T is a topology. (b) Show that if AC X is a countable subset, then the derived set A' is empty. (c) Show that if AC X is an uncountable subset, then A' = X.

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Solution a To show that T is a topology we need to check three axioms The empty set and X are in T This is true by definition since the empty set is empty and X X is empty which are both countable T i... View the full answer