NOUN POP Past Questions: MTH 401 - General Topology I

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October Examination 2019

Course Code: MTH 401

Course Title: General Topology I

Credit Unit: 3

Time Allowed: 3 Hours

Instruction: Answer Question Number One and Any Other Four Questions

1. (a) Define the following terms:

(i) condensing point (3 marks)

(ii) accumulation point of , where is a subset of of the

metric space . (4 marks)

(b) Let be a metric space and let be a subspace of . Let be subset of Y. Show that A is closed in Y if and only if there exists a set which is closed in E such that . (8 marks)

(c) Show that limits are unique in metric spaces (i.e. if converges to both and then ). (7 marks)

2. Let be defined by . Prove that is continuous at (−1, 6). (12 marks)

3. (a) Show that a subset of a metric space is closed in if and only if its complement is open in . (7 marks)

(b) Show that every singleton subset of any metric space is closed. Hence, every finite set is closed. (5 marks)

4. (a) Let and be two metric spaces and let denote their cartesian product, where is endowed with its own metric. Define Euclidean metric on . (7 marks)

(b) Show that every subsequence of a convergent sequence converges, and it converges to the same limit as does the mother sequence. (5 marks)

5. Prove that in any metric space , each open ball is an open set in . (12 marks)

6. Show that a metric space is connected if and only if the only subsets of which are both open and closed are and . (12 marks

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## NOUN POP Past Questions: MTH 401 - General Topology I

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