Years Ago Allen Shields*

In 1987 w e h a v e the f i f t ie th a n n i v e r s a r y of M. H. S tone ' s [1937] "Appl ica t ions of the theory of Boolean r ings to genera l t opo logy . " This p a p e r was very influ- ential; a m o n g the m a n y th ings tha t it conta ins are S tone ' s contr ibut ions to the Stone-Cech compactifica- t ion and to the Stone-Weierstrass theorem. (For the c o n v e n i e n c e of the r eade r , t he se concep t s are ex- p la ined briefly at the end of this article.) In reading the in t roduct ion to Stone 's paper , however , one is s truck that these concepts are hard ly ment ioned . (The same is t rue for the two reviews of this paper; by Garret t Birkhoff in the Zentralblatt, and by Rinow in the Jahr- buch; see the bibl iography.) We quote f rom Stone 's in- t roduction:

In an earlier paper (Stone [1936]) we have developed an abstract theory of Boolean algebras and their representa- tions by algebras of classes. We now relate this theory to the study of general topology.

The first part of our discussion is devoted to showing that the theory of Boolean rings is mathematically equiva- lent to the theory of locally bicompact, totally discon- nected topological spaces . . . . It is thus convenient to call the spaces corresponding . . . to Boolean rings, Boolean spaces. [Note: At that time a space was said to be compact if each infinite subset had a limit point, and bicompact if each open cover had a finite subcover (see Alexandroff and Hopf [1935], Chap. I1, pp. 84-86). Also, a Boolean ring is a ring in which each element is idempotent.]

�9 . . In the second part of our discussion . . . we pro- pose the problem of representing an arbitrary T o space by means of maps in bicompact Boolean spaces . . . .

The general mapping theory which we have outlined in the preceding paragraph is sufficiently complicated to suggest a search for simplifications . . . . [The theory] leads us at once to the consideration of a class of topo- logical spaces to which little attention has been paid in the past. These spaces are characterized by the property that in them the regular open se ts- - that is, the interiors of closed sets--constitute bases. Since they are more general than the regular spaces, we call them semi-regular spaces.

* Column editor's address: Department of Mathemat ics , Univers i ty of Michigan, A n n Arbor, MI 48109 USA

After discussing the semi-regular and regular spaces in detail, we consider the completely regular spaces. Here it is necessary for us to study the class of all bounded con- tinuous real functions . . . . We obtain a reasonably com- plete algebraic insight into the structure of this class and its correlation with the structure of the underlying topo- logical space.

The contents of the present paper may be summarized systematically under the following headings: Chapter I, Boolean spaces: w Introduction to topological concepts; w Relations between algebra and topology; w Universal Boolean rings and spaces. Chapter II, Maps in Boolean spaces: w The general theory of maps; w Construction of Boolean maps; w Relations between algebraic and other maps; w Applications to the theory of extensions; w Totally-disconnected and discrete spaces. Chapter III, Stronger separation conditions: w Semi-regular spaces; w Regular spaces; w Completely regular spaces.

Stone-(~ech Compactification. Cech ' s contr ibut ion is Cech [1937]�9 We quote f rom his introduction�9

The theory of bicompact spaces was extensively studied by P. Alexandroff and P. Urysohn [1929] . . . . An impor- tant result was added by Tychonoff [1929], who proved that complete regularity is the necessary and sufficient condition for a topological space to be a subset of some bicompact Hausdorff space. As a matter of fact, Tychonoff proves more, viz. that, given a completely regular space S, there exists a bicompact Hausdorff space [3(S) such that (i) S is dense in [3(S), (ii) any bounded continuous real func- tion defined in the domain S admits of a continuous ex- tension to the domain [3(S).

If this were correct, it would seem that an injustice had occurred, and one should call this the Tychonoff compactif icat ion. Before turning to Tychonoff ' s paper we note that Cech does himself establish the existence of [3(S) wi th the above propert ies , on pages 831-832. To do this he cons ide rs the set C~ of all con t inuous f u n c t i o n s f r o m S to the c losed un i t i n t e rva l I. He chooses a copy of I for each e lement of O and forms the car tes ian p roduc t . Then S is m a p p e d into this p roduc t (x --+ {fix)}, f ~ 9), and the comple te regu-

THE MATHEMATICAL INTELLIGENCER VOL. 9, NO. 2 �9 1987 Springer-Verlag New York 61

larity gives a sufficient supply of continuous functions so that this map is a homeomorphism onto its range. The closure of the range is denoted ~(S); it is imme- diate that each f in 9 extends to the product, and hence to ~(S). He also proves a uniqueness theorem. He goes on to study the case when S is normal, or first countable, or discrete.

Reading Tychonoff one does not find result (ii) above cited by Cech. [Note: when pronouncing Ty- chonoff, the accent is on the first syllable.] What Ty- chonoff does do, among other things, is introduce completely regular spaces and prove that they are pre- cisely the subspaces of compact Hausdorff spaces. He also proves that an arbitrary product of unit intervals is bicompact (his proof works for the product of bi- compact Hausdorff spaces in general, but he does not mention this). For an arbitrary cardinal number ,r he forms the product R~ of -r unit intervals, and shows that this space is "universal" in the sense that any completely regular space with a neighborhood basis of cardinality at most -r embeds in R~. He notes that this contains Urysohn's theorem that a normal space with a countable basis is metrisable (Urysohn [1925]). Fi- nally, he gives examples to show that the class of com- pletely regular spaces lies properly between the reg- ular spaces and the normal spaces.

For more information see the book of Gillman and Jerison [1960], especially the historical notes starting on p. 266. There is a curious question here. When and where did Tychonoff prove his theorem on the com- pactness of a product of compact spaces? Gillman and Jerison, p. 269, refer to Tychonoff [1929] and [19352], whereas Kelley [1950] refers only to Tychonoff [1935~]. Actually, neither of the 1935 papers is especially rele- vant here; the 1929 paper discussed above seems the best reference. Apparent ly the importance of the theorem was not at first recognized; for example, it is not mentioned in the book of Alexandroff and Hopf [1935], though they do give the case of two factors as an exercise for the reader (Chap. II, w Aufgabe 2, p. 86). Cech [1937] states and proves the theorem (p. 830) without referring to TychonofL though the proof is similar tovTychonoff's 1929 proof for the product of in- tervals. (Cech does refer to this paper elsewhere in his paper.)

X, then ~ = C(X). Stone [19482] gives an exposition of this and related results.

Explanatory Note. If S is a completely regular topo- logical space, then there is a compact Hausdorff space, ~(S), such that, (1) ~(S) contains S as a dense subset, (2) each bounded continuous real function on S ex- tends to be continuous on ~(S). The space ~(S) is es- sentially unique, and is called the Stone-Cech compac- tification of S. This designation is used in the Russian literature; see p. 9 of P. S. Alexandroff's introduction to the Russian edition of Kelley [1957]. Stone [19481] gives an exposition of this theory.

One other point of interest, not directly related to the above, turned up in this investigation. On p. 7 of his introduction to the Russian edition of Kelley [1957] P. S. Alexandroff mentions Moore-Smith conver- gence. The usual reference is to Moore-Smith [1922]. Alexandroff states that "this same concept was discov- ered two decades earlier by the talented Odessa math- ematician S. O. Satunovski." Unfortunately he does not give a reference. The book Mathematika v SSSR (Vol. 2, p. 766) lists ~atunovski, Samuil OsipoviG 1859-1929, Odessa. This book lists only publications since the Revolution, and there is only one. A check of the Jahrbuch shows that S. O. Scha tunowsky pub- lished several articles in the period 1895-1912, but

Stone-Weierstrass Theorem. Just over 100 years ago Weierstrass [1885] proved that every continuous func- tion on the closed unit interval is the uniform limit of polynomials. I would like to make a number of com- ments in this connection, but they will have to await another occasion.

The Stone-Weierstrass theorem states that if X is a compact Hausdorff space, if C(X) denotes the algebra of all continuous real functions on X with the su- premum norm, and if ~ is a norm closed subalgebra that contains the constants and separates the points of

62 THE MATHEMATICAL INTELLIGENCER VOL. 9, NO. 2, 1987

n o n e s e e m s r e l e v a n t here . The Matem. Entsikloped. (Vol. 3, p. 890) d i scusses c o n v e r g e n c e a long d i rec ted sets, a n d refers to M o o r e - S m i t h b u t no t to Sa tunovsk i - - h o w e v e r , the encyc loped ia does no t a t t e m p t to be his tor ical ly comple te .

A d d e d in proof : R u d i n [1973], A p p e n d i x B, p. 383, br ie f ly d i s c u s s e s the c o n t r i b u t i o n s of C e c h a n d Ty- c h o n o f f a n d c o n c l u d e s : " T h u s it a p p e a r s t ha t C e c h p r o v e d the T y c h o n o f f t h e o r e m , w h e r e a s T y c h o n o f f f o u n d the Cech c o m p a c t i f i c a t i o n - - a g o o d i l lustrat ion of the historical reliability of ma thema t i ca l n o m e n c l a - t u re . "

R e f e r e n c e s

Abbreviations: Jrb., Zbl., and MR denote, respectively: Jahr- buch fiber die Fortschritte der Mathematik, Zentralblatt ffir Mathe- matik und ihre Grenzgebiete, and Mathematical Reviews.

P. S. Alexandroff and P. S. Urysohn [1929], M6moire sur les espaces topologiques compacts, d6di6 a Monsieur D. Egoroff, Kon. Akad. Weten. Amsterdam, 14, 1-96. Jrb. 552, 960-963.

P. S. Alexandroff and H. Hopf [1935], Topologie I, Springer- Verlag, Berlin. Jrb. 611, 602-606; Zbl. 13, 79-81.

E. Cech [1937], On bicompact spaces, Ann. of Math. 38, 823-844. Jrb. 631, 570-571; Zbl. 17, 428-429.

Leonard Gillman and Meyer Jerison [1960], Rings of Contin- uous Functions, Van Nostrand, Princeton. MR 22 #6994.

John L. Kelley [1957], General Topology, Van Nost rand, Princeton. MR 16, 1136. Russian edition: Ob~aya topolo- giya, Izdat. "Nauka", Moscow (1968). MR 39 #907.

Matemati~eskaya Entsiklopediya, Izdat. "Sovet. Entsikloped.", Moscow, Vol. 1 (1977)--Vol. 5 (1985).

Matematika v SSSR za sorok let (Mathematics in the USSR for fo r ty years ) 1917-1957, Gos. Izda t . Fiz.-Mat. Lit. , Moscow, 1959.

E. H. Moore and H. L. Smith [1922], A general theory of limits, Amer. J. Math. 44 (1922), 102-121. Jrb. 48, 1254.

W. Rudin [1973], Functional Analysis, McGraw-Hill, New York.

M. H. Stone [1936], The theory of representat ions for Boolean algebras, Trans. Amer. Math. Soc. 40, 37-111. Jrb. 62~, 33-34; Zbl. 14, 340.

- - [ 1 9 3 7 ] , Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41, 375-481. Jrb. 63~, 1173; Zbl. 17, 135.

- - [ 1 9 4 8 1 ] , On the compactification of topological spaces, Ann. Soc. Polon. Math. 21, 153-160. MR 10, 137.

- - [ 1 9 4 8 2 ] , The generalized Weierstrass approximation theorem, Math. Mag. 21., 167-184, 237-254. MR 10, 255.

A. N. Tychonoff [1929], Uber die topologische Erweiterung von R6umen,.Math. Ann. 102, 544-561. Jrb. 552, 963.

- - [ 1 9 3 5 1 ] , Uber einen Funktionenraum, Math. Ann. 111, 762-766. Jrb. 612, 1194-1195; Zbl. 12, 308.

- - [ 1 9 3 5 2 ] , Ein Fixpunktsatz, Math. Ann. 111, 767-776, Jrb. 612, 1195; Zbl. 12, 308.

P. S. Urysohn [1925], Zum Metrisationsproblem, Math. Ann. 94, 309-315. Jrb. 51, 453.

K. Weierstrass [1885], Uber die analytische Darstellbarkeit sogenannter willktirlicher Functionen einer reellen Ver~in- derlichen, Sitz, Berichte K6n. Preuss. Akad. Wiss., Berlin (1885), 633-639 and 789-805. Jrb. 17 (1885), 384-388.

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THE MATHEMATICAL INTELLIGENCER VOL. 9, NO. 2, 1987 63