Appendix
Transfinite Induction and General
Topology
In order to help make our discussions relatively self-contained, it will be convenient to collect together in this Appendix the basic facts and notation we
need from the theory of ordinals
1 and from the subject of general topology.
A.1 The Principle of Transfinite Induction
We begin with a brief discussion of the theory of ordinals and transfinite
induction. In particular, we give a statement of the principle of transfinite
induction in the form in which we make use of it on a number of occasions.
A.1.1 Definition A partially ordered set X is well-ordered or is a wellordering if each non-empty subset of X has a first or least element.
A.1.2 Example (1) The set N of natural numbers is well-ordered in the usual
ordering ≤ on N.
(2) The set Z of integers is not well-ordered in the usual ordering ≤ on Z.
A.1.3 Lemma The following statements hold.
(a) Every well-ordered set is linearly ordered.
(b) No well-ordered set contains an infinite strictly descending sequence.
Proof: (a) Let (X, ≤ X ) be a well-ordered set, and let x, y ∈ X. Then the set
{x, y} is a non-empty subset of X and hence has a least element, x, say. But
then x ≤ X y, which establishes (a).
For (b), suppose that (x n ) n∈N is an infinite strictly decreasing sequence in
the well-ordered set (X, ≤ X ). Then {x n | n ∈ N} itself is a non-empty subset
1 Our treatment of these matters is informal and non-axiomatic and is in the spirit of the
book [Halmos, 1998] to which we refer the reader for further details.
229
Transfinite Induction and General
Topology
In order to help make our discussions relatively self-contained, it will be convenient to collect together in this Appendix the basic facts and notation we
need from the theory of ordinals
1 and from the subject of general topology.
A.1 The Principle of Transfinite Induction
We begin with a brief discussion of the theory of ordinals and transfinite
induction. In particular, we give a statement of the principle of transfinite
induction in the form in which we make use of it on a number of occasions.
A.1.1 Definition A partially ordered set X is well-ordered or is a wellordering if each non-empty subset of X has a first or least element.
A.1.2 Example (1) The set N of natural numbers is well-ordered in the usual
ordering ≤ on N.
(2) The set Z of integers is not well-ordered in the usual ordering ≤ on Z.
A.1.3 Lemma The following statements hold.
(a) Every well-ordered set is linearly ordered.
(b) No well-ordered set contains an infinite strictly descending sequence.
Proof: (a) Let (X, ≤ X ) be a well-ordered set, and let x, y ∈ X. Then the set
{x, y} is a non-empty subset of X and hence has a least element, x, say. But
then x ≤ X y, which establishes (a).
For (b), suppose that (x n ) n∈N is an infinite strictly decreasing sequence in
the well-ordered set (X, ≤ X ). Then {x n | n ∈ N} itself is a non-empty subset
1 Our treatment of these matters is informal and non-axiomatic and is in the spirit of the
book [Halmos, 1998] to which we refer the reader for further details.
229