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ASSIGENMENT II B.SC.II UNIT I

ASSIGENMENT II B.SC.II UNIT I. TOPOLOGY OF REAL NUMBERS. Prove that arbitrary union of open sets is open. Prove that the set of rationals is not order complete. Prove that a set A is compact iff every open cover of A has finite subcover .

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ASSIGENMENT II B.SC.II UNIT I

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  1. ASSIGENMENT II B.SC.II UNIT I TOPOLOGY OF REAL NUMBERS

  2. Prove that arbitrary union of open sets is open. • Prove that the set of rationals is not order complete. • Prove that a set A is compact iff every open cover of A has finite subcover. • Set A is infinite bounded subset of R, then prove that least upper bound of A either belongs to A or is a limit point of A. • Prove that the closure of a set AR is the smallest closed superset of A.

  3. 6. Prove that the interior of set A is the largest open subset of A. 7. Prove that every non-empty set of real numbers which is bounded below has glb. 8. Prove that every infinite bounded subset of real numbers has a limit point. 9. Prove that the supremum and infimum of a set A are also supremum and infimum of A and are contained in  A according as A is bounded above or below. 10. Prove that the intersection of an arbitrary family of closed sets is closed.

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