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TON DUC THANG UNIVERSITY
FACULTY OF MATHEMATICS AND STATISTICS HW 2 Recall:
 A set A is closed if its complement is open.
 In a topological space X,  and X are closed.
Intersection of closed sets is closed.
Finite union of closed sets is closed.
 A neighborhood of a point x is an open set that contains x. The set of all neighborhoods of x is denoted by Nx .
 A set A is open iff  x  A,  V  Nx , V  A. o o
 The interior of A, denoted by A is the set of all interior points of A, i.e. A = {x :  V Nx , V  A}.
 Let A be a subset of a topological space X. Then o o
A  A and A is open. o
A is the largest open set which is contained in A. o
A is open iff A = A. o o o A = A .
 In R, every closed interval is closed. Problems:
1/ Suppose A is closed and B is open. Prove that A\B is closed and B\A is open. o
2/ Suppose B is open and B  A. Prove that B  A . o o
3/ Let A, B  X. Prove that if A  B then A  B .
4/ Let A, B  X. Prove that: o o o o a) o A B  (A  B) . b) o A B  (A  B) .
5/ Let Ai  X, i  I. Prove that: o o o   o   a) A   A . b) A   A . i i  i i  i  i  i  i  Note. Here, o
(A  B) = the interior of AB. INTRODUCTION TO TOPOLOGY Dr. Chu Duc Khanh Semester 2, 2025-2026 TON DUC THANG UNIVERSITY
FACULTY OF MATHEMATICS AND STATISTICS o 6/ In R, find A where i) A = (0,2] ii) A = [1,3) iii) A = [–2,5] iv) A = (–,0] v) A = (–2,+) vi) A = (–3,6) vii) A = (–,3) viii) A = (–1,+) ix) A = N x) A = Z xi) A = Q xii) A = R\Q o
7/ Let A be a finite subset of R. Prove that A = .
8/ Prove that any finite subset of R is closed. o
9/ Is the following assertion true: X\ A  X\A. *** INTRODUCTION TO TOPOLOGY Dr. Chu Duc Khanh Semester 2, 2025-2026