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Noun (National open university) Live 2019 POP Exam questions
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[QUOTE="Bracet, post: 3630, member: 477"] NATIONAL OPEN UNIVERSITY OF NIGERIA Plot 91, Cadastral Zone, Nnamdi Azikiwe Expressway, Jabi, Abuja. FACULTY OF SCIENCES April/May Examination 2019 Course Code: MTH401 Course Title: General Topology 1 Credit Unit: 3 Time allowed: 3 HOURS Total: 70 Marks Instruction: ATTEMPT NUMBER ONE (1) AND ANY OTHER FOUR (4) QUESTIONS 1. (a) Define a metric space (4marks) (b) Let denote the set of real numbers and let be defined by ( ) | | for all Show that is a metric on (6marks) (c) State Triangle and H ̈lder’s inequalities (4marks) (d) State and prove Minkowski’s inequality. (8marks) 2. (a) Define the following: (i) Open ball (ii) Closed ball (iii) Spheres. (5marks) (b) Let be endowed with the Euclidean metric. ( ) Σ*( ) + ⁄ for all ( ) ( ) Describe the following sets(i) (( ) )(ii) ̅(( ) )(iii) (( ) ) where ( ) (iv) ( ) for arbitrary (7marks) 3. (a) Define the closure of a set . (5marks) (b) Every singleton subset of any metric space is closed. Hence, every finite set is closed. (7marks) 4. (a) Let * + be a sequence of points in a metric space ( ) When is a point said to be a limit point of the sequence * + (5marks) (b) Show that * +converges to in , if and only if * ( )+ converges to in (7marks) 5. (a) When is a sequence said to be a Cauchy in a metric space? (5marks) (b) Prove that every convergent sequence in a metric space is Cauchy. (7marks) 6. (a) Define a connected space (5marks) (b) Prove that the image of a connected space under a continuous map is connected. (7marks) [/QUOTE]
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