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TON DUC THANG UNIVERSITY
FACULTY OF MATHEMATICS AND STATISTICS
INTRODUCTION TO TOPOLOGY
Semester 2, 2025-2026
Dr. Chu Duc Khanh
HW 1
Recall:
Let f: X Y be a map, A X, B Y.
The image of A by f is the set f(A) = {f(x) : x A}
The inverse image of B by f is the set
1
f (B)
= {x X : f(x) B}
Therefore,
o y f(A) iff x A, f(x) = y
o x
1
f (B)
iff f(x) B
A topology on a set X is a collection of subsets of X (that called open sets) such that:
(i) and X are open.
(ii) Union of open sets is open.
(iii) Finite intersection of open sets is open.
The usual topology on R is defined as follows:
A subset A of R is open iff x A, open interval (,) such that x (,) A.
In R, every open interval is open.
Problems:
1/ Let f: X Y and A
1
, A
2
X.
Prove that if A
1
A
2
then f(A
1
) f(A
2
).
2/ Let f: X Y and B
1
, B
2
Y
Prove that if B
1
B
2
then f
–1
(B
1
) f
–1
(B
2
).
3/ Let f: X Y and A
1
, A
2
X.
Prove that f(A
1
A
2
) = f(A
1
) f(A
2
) and f(A
1
A
2
) f(A
1
)f(A
2
).
4/ Let f: X Y and B
1
, B
2
Y
Prove that f
–1
(B
1
B
2
) = f
–1
(B
1
)f
–1
(B
2
) and f
–1
(B
1
B
2
) = f
–1
(B
1
)f
–1
(B
2
).
5/ Let f: X Y and A X. Prove that A
1
f (f(A))
.
6/ Let f: X Y and B Y. Prove that
1
f(f (B)) B
.
TON DUC THANG UNIVERSITY
FACULTY OF MATHEMATICS AND STATISTICS
INTRODUCTION TO TOPOLOGY
Semester 2, 2025-2026
Dr. Chu Duc Khanh
7/ Let f: X Y and A
i
X, i I. Prove that:
a/
ii
i I i I
f A f(A )


.
b/
ii
i I i I
f A f(A )


.
8/ Let f: X Y and B
i
Y, i I. Prove that:
a/
.
b/
11
ii
i I i I
f B f (B )


.
9/ Let f: X Y and B Y. Prove that
11
f (Y \ B) X \f (B)

.
10/ For each n N, put
n
1
E ,2
n


. Prove that
n
n1
E (0,2)
.
11/ For each n N, put
n
1
F 1 ,2
n



. Prove that
n
n1
F [1,2]
.
12/ In R, prove that:
a/ A = (–2,1) (5,8) is open.
b/ R\Z is open.
c/ B = [0,1) is not open.
d/ Q is not open.
13/ Let O
1
and O
2
be two topologies on a set X. Prove that = O
1
O
2
is a topology on X.
14/ Let O
j
(j J) be a family of topologies on a set X. Prove that
j
jJ
 O
is a topology on X.
***

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TON DUC THANG UNIVERSITY
FACULTY OF MATHEMATICS AND STATISTICS HW 1 Recall:
 Let f: X  Y be a map, A  X, B  Y.
The image of A by f is the set f(A) = {f(x) : x  A} 
The inverse image of B by f is the set 1
f (B) = {x  X : f(x)  B} Therefore,
o y  f(A) iff x A, f(x) = y o x  1  f (B) iff f(x)  B
 A topology on a set X is a collection of subsets of X (that called open sets) such that: (i)  and X are open.
(ii) Union of open sets is open.
(iii) Finite intersection of open sets is open.
 The usual topology on R is defined as follows:
A subset A of R is open iff x  A,  open interval (,) such that x  (,)  A.
 In R, every open interval is open. Problems:
1/ Let f: X  Y and A1, A2  X.
Prove that if A1  A2 then f(A1)  f(A2).
2/ Let f: X  Y and B1, B2  Y
Prove that if B1  B2 then f –1(B1)  f –1(B2).
3/ Let f: X  Y and A1, A2  X. Prove that f(A 
1  A2) = f(A1)  f(A2) and f(A1 A2)  f(A1)f(A2).
4/ Let f: X  Y and B1, B2  Y Prove that f –1(B   1
B2) = f –1(B1)f –1(B2) and f –1(B1 B2) = f –1(B1)f –1(B2). 
5/ Let f: X  Y and A  X. Prove that A  1 f (f (A)) . 
6/ Let f: X  Y and B  Y. Prove that 1 f (f (B))  B . INTRODUCTION TO TOPOLOGY Dr. Chu Duc Khanh Semester 2, 2025-2026 TON DUC THANG UNIVERSITY
FACULTY OF MATHEMATICS AND STATISTICS
7/ Let f: X  Y and Ai  X, i  I. Prove that:   a/ f  A   f (A ) . i i  i I   i I    b/ f  A   f (A ) . i i  i I   i I 
8/ Let f: X  Y and Bi  Y, i  I. Prove that:     a/ 1 1 f  B   f (B ) . i i  i I   i I      b/ 1 1 f  B   f (B ) . i i  i I   i I   
9/ Let f: X  Y and B  Y. Prove that 1 1 f (Y \ B)  X \ f (B) .   1 
10/ For each n  N, put E  , 2 . Prove that E  (0, 2) . n    n  n n 1    1 
11/ For each n  N, put F  1 , 2 . Prove that F  [1, 2] . n    n  n n 1  12/ In R, prove that:
a/ A = (–2,1)  (5,8) is open. b/ R\Z is open. c/ B = [0,1) is not open. d/ Q is not open.
13/ Let O1 and O2 be two topologies on a set X. Prove that  = O1  O2 is a topology on X. 14/ Let O  
j (j  J) be a family of topologies on a set X. Prove that O is a topology on X. j j J  *** INTRODUCTION TO TOPOLOGY Dr. Chu Duc Khanh Semester 2, 2025-2026

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