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    Closure of a set in a a topoogical subspace

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    Let Y be a subspace of X and let A be a subset of Y. Denote by Cl(A_X) the closure of A in the topological space X and by Cl(A_Y) the closure of A in the topological space Y. Prove that Cl(A_Y) is a subset of Cl(A_X) . Show that in general Cl(A_Y) not equal Cl(A_X). See the attached file.

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    Solution Summary

    A proof of the fact that the closure of a set in a topological subspace is a subset of the cosure of the same set in a bigger space is given. A counterexample to the equality claim is given. The solution is in a PDFfile.