Let X be a topological space and let AcX Then The interior o

Let X be a topological space and let AcX. Then The interior of A, denoted A, is the union of all open sets contained in A The closure of A, denoted A, is the intersection of all closed sets containing A. The boundary of A, denoted A, is the intersection Anx-A Observe that ACACA

Solution

As A is contained in X,

Consider all open sets contained in A

Then union of all open sets contained in A denoted by A0, is always the interior of A

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If A\' denote a closed set in A

Consider the intersection al all such closed sets

Then they will end up in the boundary of set of A

Hence closure of A is the intersection of all closed sets containing A

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The boundary of A is the common line between A and A completment.

Or A and X-A

Hence A\' the closed set of A and (X-A)\' the closed set of complement of A intersect in the boundary.