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Math 400: Topology I Question Paper

Math 400: Topology I

Course:Bachelor Of Science In Mathematics

Institution: Chuka University question papers

Exam Year:2013

CHUKA

UNIVERSITY

UNIVERSITY EXAMINATIONS

FOURTH YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF SCIENCE (IN MATHEMATICS)

MATH 400: TOPOLOGY I

STREAMS: BSc. (In Mathematics) Y4S1 TIME: 2 HOURS

DAY/DATE: THURSDAY 8/8/2013 11.30 A.M. – 1.30 P.M.
INSTRUCTIONS:

Answer Question 1 (Compulsory) an any other Two questions.
Adhere to the instructions on the answer booklet.
Do NOT write on the question paper.

QUESTION 1 (compulsory) – (30 Marks)

(i) Given three sets X, Y and Z, show that

Z\(X?Y)=(Z\X)n(Z\Y) (3 marks]

(ii) Hence prove that

X\?A?= n(X\A?),??A [3 marks]

Let (X,t) be a topological space, show that

X,?are closed sets. [2 marks]
A finite union of closed sets is also closed. [3 marks]

Let X = {a, b, c, d, f} and

Y = {?,X, {b, c}, {a}, {a, b, c}, {f},{a, f}, {b, c, f}, {a, b, c, f}}

Let A = {a, c, d}

Show that a is not a limit point of A whereas b is a limit point of A. [5 marks]
Consider the following topology on X = {a, b, c, d, e}.

Let t = {X,?, {a}, {a, b}, {a, c, d}, {a, b, c, d}, {a, b, e}}

Suppose A = {a, b, c}. Find

the interior of A [2 marks]
the exterior of A [2 marks]
the boundary of A [2 marks]

Prove that a point p?X is an accumulation point of A?X if and only if every member of some local base ß_p at the point p contains a point of A different from p. [4 marks]

Let f:X?Y and g:Y?Z be continuous functions, prove that the composite is also continous. [4 marks]

QUESTION 2 : (20 MARKS)

Define the following terms as used in topology.

A discrete topology [1 mark]
A Sierpinski’s topology [1 mark]
A nowhere dense subset A?X [1 mark]
A base for the topology t [2 marks]
A topological property (P) [1 mark]

Let X be a topological space and A,B?X. Denote A'' a derived set of A. Show that

?(A?B)?^''=A''?B'' [5 marks]
(A^'' )^''=A'' [4 marks]

Let (X,t) be a topological space and A?X, prove that the interior of (i.e A°) is the union of all open subsets of A. [5 marks]

QUESTION 3 : (20 MARKS)

Let ß be a class of subsets of a non-empty set X. Prove that ß is a base for some topology on X if and only if

X= ?{B:B?ß}

For anyB,B^* ?ß,BnB^* is the union of members of ß or equivalently if p?BnB^* then B_P ?ß such that p?B_p? BnB^*. [11 marks]

Let f: X_1?X_2 where X_1=X_2={0,1} and are such that (X_1,D) and
(X_2, $) defined by f(1)=1
f(o)=0
(where D is a discrete topology and $ is Sierpinski’s topology)
Show that f is not continous but f^(-1) is continous. [5 marks]

(c) Distinguish a regular space and a Normal space. [4 marks]

QUESTION 4 : (20 MARKS)

Let (X,t_x)and (Y, t_y) be topological spaces and f:X?Y be a map, then prove that the following statements are equivalent.

fis an open map.
f(A°)?(f(A) )°
If ß is a base for t_x then f(B) is t_x – open for every B?ß.
?x?X and any neighbourhood N_x of X a neighbourhood N_f (x) in Y such that
f(x)?N_f (x)? f(N_x ). [10 marks]

Let (X,t_x) and Y,t_y)be topological spaces, show that f:X?Y is closed if and only if
¯(f(A))? f(¯A),?A?X. [6 marks]

Let X = {a, b, c, d} and
t_x = {{a, b}, {a}, {b}, X, Ø}

andY={x,y,z,t} with
t_y = {{x}, {y}, {x,y}, Y, Ø}

Define the function f as

Show that f is a homoemorphism. [4 marks]
QUESTION 5: (20 MARKS)

Prove that a topological space X is a t_1-space if and only if every simpleton subset {p} ?X is closed. [8 marks]

Show that every metric space (X, d) is a hausdorff space. [6 marks]

Using appropriate examples illustrate the fact that T1 space ? To space and
T2space?T1space but the reverse of this is not true ie.To? T1 and T1?T2.
[6 marks]

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Math 400: Topology I Question Paper

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