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Problems in the Point-Set Topology of R^n

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Let Fr(A) denote the frontier set of A and Cl(A) denote the closure of A, where A is a subset of R^n. Solve the following problems.

Exercise 2.6: For any set A, Fr(A) is closed.

Exercise 2.12: If A and B are any sets, prove that Cl(A and B) belongs to Cl(A) and Cl(B). Give an example where Cl(A and B) is empty, but Cl(A) and Cl(B) is not empty.

Exercise 2.17: Show that x is the only limit point of the sequence constructed in Theorem 2.11.

Exercise 2.25: Prove that the composition of two continuous functions is continuous.

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Please find the solutions to the stated problems attached.

Basic Facts from Topology

Here we state some basic definitions and properties that are required to do the problems. ...

Solution Summary

This solution provides a step-by-step process and accompanying explanation for solving four basic problems in the point-set topology of R^n. These problems are related to the properties of frontier sets, the closure of sets, limit points, and the composition of continuous functions.