By Boyer Ch. P.

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**Example text**

The following theorem states a condition that is often used as the definition of a regular space. 2-3] Ta- AND T4-SPACES 41 THEOREM 2-4. A T 1-space Sis regular if and only if for each point p inS and each open set U containing p, there is an open set V containing p whose closure V is contained in U. Proof: If S is regular and the point p lies in an open set U, then Axiom T 3 states that there exist disjoint open sets V and W, with pin V, and W containing the closed set S - U. Since V n (S - U) is empty, V lies in U.

THEOREM 1-27. Every set can be well-ordered. We do not prove this. We merely observe that this theorem means that into every set a simple-order relation can be introduced, using the axiom of choice, in such a way that the set is well-ordered under this order relation. The real numbers are not well-ordered by size; there is no smallest positive number, for example. But by Theorem 1-27, there exists an order relation in which the reals are well-ordered although no such relation has ever been explicitly defined.

The sets {Ox} coverS, and hence some finite number 0:~: 1 , • • • , O:en covers S. But then there are at most n points in X. It follows that any infinite subset must have a limit point. D Suppose that Sis a compact space and that f:S --t Tis a continuous mapping of S onto a space T. If {0.. )} are nonempty and coverS. ,)}, i = 1, ... , n, covers S. Then f[f- 1 (0.. ;)] = 0 .. ; covers T, and T is also compact. 20 TOPOLOGICAL SPACES AND FUNCTIONS (CHAP. 1 If S is countably compact, and f:S ~ T is continuous and onto, consider an infinite subset X of T.