Introductory Topology

Exercises and Solutions

Mohammed Hichem Mortad

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789813146952
• Edition: 2
• Topic: Mathématiques
• Subtopic: Mathématiques appliquées

The book offers a good introduction to topology through solved exercises. It is mainly intended for undergraduate students. Most exercises are given with detailed solutions.In the second edition, some significant changes have been made, other than the additional exercises. There are also additional proofs (as exercises) of many results in the old section "What You Need To Know", which has been improved and renamed in the new edition as "Essential Background". Indeed, it has been considerably beefed up as it now includes more remarks and results for readers' convenience. The interesting sections "True or False" and "Tests" have remained as they were, apart from a very few changes.

Part 1

Exercises

CHAPTER 1

General Notions: Sets, Functions et al.

1.1. Essential Background

In this section we recall some of the basic material on sets, functions, countability as well as consequences of the least upper bounded property. Other basic concepts will be assumed to be known by the reader.

1.1.1. Sets. We start with the definition of a set. We just adopt the naive definition of it as many textbooks do. We shall not go into much detail of Set Theory. It can be a quite complicated theory and a word from J. M. Møller suffices as a warning. He said in [18]: "We don’t really know what a set is but neither do the biologists know what life is and that doesn’t stop them from investigating it". So, we do advise the student to be careful when dealing with set theory in order to avoid contradictions and paradoxes. Before commencing, we give just one illustrative paradox which is the Russel’s set (from Russel’s paradox: if someone says "I’m lying", is he lying? ): Let

R = {A : A ∉ A}

be the set of all sets which are not an element of itself. So is R ∈ R? Or is it not?

DEFINITION 1.1.1. A set is a well-defined collection of objects or elements. We denote sets in general by capital letters: A, B, C; X, Y , Z, etc...Elements or objects belonging to the set are usually designated by lowercase letters: a, b, c; x, y, z, etc...

If a is an element of A, then we write: a ∈ A. Otherwise, we write that a ∉ A.

If a set does not contain any element, then it is called the empty (or void) set, and it is denoted by: ∅ or {} (J.B. Conway in [7] likes to denote it by □).

EXAMPLES 1.1.2.

(1){−4, 1, 2} is a set constituted of the elements 1, 2 and −4.

(2){x : x ≥ 0} is also a set but we cannot write down all its elements.

One element sets (e.g. {2}) have a particular name.

DEFINITION 1.1.3. A singleton is a set with exactly one element.

The notion of a "smaller" or "larger" set cannot be defined rigourously even though we will be saying it from time to time. We have a substitute.

DEFINITION 1.1.4. Let A and B be two sets.

(1)We say that A is a subset of B (or that B is a superset of A) if every element of A is an element of B, and we write

A ⊂ B or B ⊃ A.

We may also say that A is contained (or included) in B (or that B contains (or includes) A). The relation "⊂" is called inclusion.

(2)The two sets A and B are equal, and we write A = B, if

A ⊂ B and B ⊂ A.

Otherwise, we write A ≠ B.

(3)We say that A is a proper subset of B if A ⊂ B with A ≠ B. We denote this by A ⊊ B, and we call "⊊" a proper inclusion.

EXAMPLE 1.1.5. W...