By S. M. Srivastava (auth.)

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The "only if" part is trivially seen. 11 Induction on Trees 29 e Let T be a tree on a well-ordered set (A, ~). We define an ordering

For any A ~ X, cl(A) will denote the intersection of all closed sets containing A. Thus cl( A) is the smallest closed set containing A and is called the closure of A. Note that D ~ X is dense if and only if cl(D) = X. The largest open set contained in A, denoted by int(A), will be called the interior of A. A set A such that x E int(A) is called a neighborhood of x. 10 For any A ~ X, X a topological space, show that X \ cl(A) = int(X \ A). Let (X,d) be a metric space, (xn) a sequence in X, and x E X.

There are many. But we shall have to wait to see an example of an uncountable well-ordered set. Another very natural question is the following: Can every set be well-ordered? In particular, can IR be well-ordered? Recall that (using AC) every set can be linearly ordered and every countable set can be well-ordered. This brings us to another very useful and equivalent form of AC. Well-Ordering Principle (WOP) Every set can be well-ordered. • Let {Ai: i E I} be a family of nonempty sets and A = Ui Ai.