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let F c X be another open set. Show that F n G is also open. Give an example where the intersection of an arbitrarily family of open sets may fails to open. please do help

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I don't have your definition of G. (Assuming G c X is an open set.) You can directly apply the definition in this case. For the case of an intersection of a family of open sets, consider the family of open sets A where An = (-infinity, 1/n) and the family of open sets B where Bn = (-1/n, infinity). The intersection of all A and B is {0}, which is closed.
Visually, consider the intersection of the X-axis and Y-axis

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