Definition 2.21. Let X be a space. Let A be a subset of X. We say that a point x € X is a limit point of A iff for every open set U in X containing x, (U \ {x}) nA # 0. The set of limit points of A is denoted A'. Exercise 2.22. Let E CN be the set of even natural numbers. Give N the Ta topology (see Exercise 2.12). What is E'? What is E?
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Solve 2.23
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Given and where is the set of all even natural numbers.
Here
Let .
Claim: is not a limit point of .
Consider .
Then and hence is a open set of which contains .
But then,
Hence we have found a open set of containing such that .
Therefore, it follows that is not a limit point of .
Since is arbitrary it follows that no natural number is a limit point of .
Therefore,
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