By John W. Tukey

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

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8 Any union of open 1et1 11 an opeh ,et. We shall be really interested in open sets only when there are "enough" of them. 9 Le•••• For a epace X, the following condition, are equlwa• lent: 6. If The open 1et1 contAlnlng x for• a nbd ba1l1 at 1. 11 ... 10. 10) every nbd of x contains x, this mapping converges to x; hence xeH implies xeH; hence H:iH. 11 is proved. 11. Let N be a nbd or x, then xeX='H~X-N", so that xeN. 10 holds, A space satisfying these conditions is a T-space. 10 is 6, 12 N is a nbd of x if and only if xeUc:N where U Is open.

It is clear (3,2) that f(plP) describes the topology of X. tt If X Is neither finite or countably Infinite, then function from P to 21 determines a space. n2! • Some readers may be interested in proving these statements and in proving that starting from a set containing a finite number, n, of distinguishable points we may construct the following numbers of spaces with the following properties. d compact). 6. Compactification. 160) that compactness is a completeness property. It is natural, therefore, to try to compactify spaces by "completing" them.

In §2 we discuss the applicability of the term "convergence" and point out the phalanx as particularly useful, In § 3 we establish the equivalence of closure, convergence and neighborhoods under very general conditions. In 14 we discuss the effectiveness or the various directed systems as carriers of convergence. In§§5 and 6 we discuss relativization and open sets, In §§7 and 8 we deal briefly with continuity and the separation axioms. In§9 we make some historical remarks, The results of this chapter may be summarized as follows: The term "convergent" concerns functions defined on directed systems.