BSc (Hons) Mathematics
Cohorts: BM/08/FT & BM/07/FT
Examinations for 2010-2011 / Semester 1
Examinations for 2009-2010 / Semester 1
MODULE: Topology
MODULE CODE: MATH 3125
Duration: 2 Hours 30 mins
Instructions to Candidates:
1. Answer any four questions.
2. Questions may be answered in any order but your answers must show
the question number clearly.
3. Always start a new question on a fresh page.
4. All questions carry equal marks.
5. Total marks 100.
This question paper contains 6 questions and 4 pages.
Page 1 of 4
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ANSWER any 4 QUESTIONS
1.(a)
Define a metric space.
Let Rn denote the set of all n-tuples of real numbers , d : Rn x Rn → R be
defined as
d ( x, y) = max {/xi –yi /}
where x= (x1 , x2,…, xn ) and y = (y1 , y2, …, yn).
Prove that d forms a metric on Rn.
(11 marks)
(b)
Sketch the open sphere centred at origin and radius 2 for the following metric on
R2:
d ( x, y) = max {/xi –yi /}
where x= (x1 , x2 ) and y = (y1 , y2 ).
(7 marks)
(c )
Let <X, d> be a metric space and x , y and z  X.
Prove that |d(x , y) – d(y , z) | ≤ d( x , z) .
(7 marks)
2.(a)
Let <X, d> be a metric space. Define
(i) an open sphere in X
(ii) an open set in X.
Prove that every open sphere in X is an open set.
(10 marks)
(b)
Let X be a metric space. Prove that finite intersection of open sets in X is open.
(8 marks)
(c)
Prove that in a discrete metric space every set is an open set.
(7 marks)
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3 (a)
What is meant by an interior point of a set A in a metric space X?
Prove that the interior of a set A (A0)
(i)
is an open set in X
(ii)
contains all open subsets of A.
(10 marks)
(b)
Let <X, d> be metric space and A, B  X.
Prove that A0  B0  (A  B) 0.
By giving an example show that the converse need not be true.
(8 marks)
(c)
Let <X, d> be metric space. Show that the complement of a singleton set in X is
an open set.
(7 marks)
4.(a)
Define a limit point in a metric space.
If A is a set in a metric space X, prove that A is the intersection of all closed
supersets of A.
(9 marks)
(b)
Let <X, d> be metric space and A  X.
A is said to be closed if it contains all its limit points. Prove that A is closed if and
only if Ac is open.
(9 marks)
(c)
Define a topology on a set X. Prove that the intersection of two topologies on X is
a topology on X.
(7 marks)
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5.(a)
Define a convergent sequence in a metric space. What can be said about the limit
of a sequence? Justify your answer.
(10 marks)
(b)
Let <X, d> , <Y, d1> and <Z, d2> be metric spaces. If f : X → Y and
g : Y → Z are continuous, prove that gf : X → Z is continuous.
(7 marks)
(c)
Let X be a set. Define a basis B for a topology τ on X.
Show that τ equals the collection of all unions of elements of B.
(8 marks)
6.(a)
Let X and Y be topological spaces and f : X → Y . Prove that the following are
equivalent:
(i) f is continuous
(ii) For every subset A of X, f( A )  f (A)
(iii) For every closed set B of Y, f -1 (B) is closed in X.
(16 marks)
(b)
Define a compact space. Prove also that the image of a compact space under a
continuous map is compact.
(9 marks)
***END OF EXAM PAPER***
Page 4 of 4
2010/S1