By John W. Tukey

The description for this e-book, Convergence and Uniformity in Topology. (AM-2), could be forthcoming.

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**Extra info for Convergence and uniformity in topology**

**Example text**

5) xeX-(X-H) = H. So we have proved 3,8. ,t,. ' is residual in a ; that is, there is an a' such that a> a' implies ae* a' implies x(a)eN, Let a• be cofinal in d. If X-x(Q•) were a nbd of x; then, for some a' , a> a• would imply x(a)eX-x(
*

A If IAI ~ IDI , converging In X • X are equiva- X, X, X Is convergent In X Is convergent In X Is convergent In every A-phalanx In X has a X. X• X. 10. If {H 0 1 C} has the property t'hat, for all 'Y c:C, n{ H0 I ce -yJ I= f. 'roof. 6 implies 4,7, Thus to prove all thesA conditions equivalent we need only show that 4. 7 implies 4. ch follows immediately from 3, 15 and 3,10. 10 are equivalent. 10. 10, and let x(-ylC) be any phalanx.. Let B =r , and let Hb = H-, • {:x:(~ 1 )I '}' 1:,')'}, where b =')' .

3,23 N la a nbd 1 of x laplle1 I la a nbd 1 of JI• 4. Effectiveness. We now inquire about the effectiveness or different directed systems as carriers or convergent objects. 16), where, since convergence is derivable from nbds all directed systems are equally favored. kl ... 1 Definition. 2 Theore11. (in the 11nse of 11-3). Proof. If a> 'B , there exist functions a(bl1'3) and b(alel) such that a> a(b) implies b(a) > b. 9), for some b' , b > b' implies x(b)eN. 9) x(ala) converges to x. If a is as effect!