Factoring functions on Cartesian products

Authors:
N. Noble and Milton Ulmer

Journal:
Trans. Amer. Math. Soc. **163** (1972), 329-339

MSC:
Primary 54.25

DOI:
https://doi.org/10.1090/S0002-9947-1972-0288721-2

MathSciNet review:
0288721

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A function on a product space is said to depend on countably many coordinates if it can be written as a function defined on some countable subproduct composed with the projection onto that subproduct. It is shown, for a completely regular Hausdorff space having uncountably many nontrivial factors, that each continuous real-valued function on depends on countably many coordinates if and only if is pseudo- -compact. It is also shown that a product space is pseudo- -compact if and only if each of its finite subproducts is. (This fact derives from a more general theorem which also shows, for example, that a product satisfies the countable chain condition if and only if each of its finite subproducts does.) All of these results are generalized in various ways.

**[A]**. B. A. Anderson,*Topologies comparable to metric topologies*, Topology Conference, Arizona State University, Tempe, Ariz., 1967, pp. 15-21.**[A]**. S. Armentrout,*A Moore space on which every real-valued continuous function is constant*, Proc. Amer. Math. Soc.**12**(1961), 106-109. MR**22**#11365. MR**0120615 (22:11365)****[C]**. W. W. Comfort,*A nonpseudocompact product space whose finite subproducts are pseudocompact*, Math. Ann.**170**(1967), 41-44. MR**35**#965. MR**0210070 (35:965)****[C]**. -,*Theory of cardinal invariants*, General Topology and its Applications, Springer-Verlag (to appear).**[C]**. H. H. Corson,*Normality in subsets of product spaces*, Amer. J. Math.**81**(1959), 785-796. MR**21**#5947. MR**0107222 (21:5947)****[CN]**. W. W. Comfort and S. Negrepontis,*Ultrafilters and the Stone-Čech compactification*(to appear).**[D]**. R. O. Davies,*An intersection theorem of Erdös and Rado*, Proc. Cambridge Philos. Soc.**63**(1967), 995-996. MR**35**#6570. MR**0215735 (35:6570)****[E]**. R. Engelking,*On functions defined on Cartesian products*, Fund. Math.**59**(1966), 221-231. MR**34**#3546. MR**0203697 (34:3546)****[ER]**. P. Erdös and R. Rado,*Intersection theorems for systems of sets*, J. London Math. Soc.**35**(1960), 85-90. MR**22**#2554. MR**0111692 (22:2554)****[F]**. Z. Frolík,*On two problems of W. W. Comfort*, Comment. Math. Univ. Carolinae**8**(1967), 139-144. MR**35**#966. MR**0210071 (35:966)****[G]**. I. Glicksberg,*Stone-Čech compactifications of products*, Trans. Amer. Math. Soc.**90**(1959), 369-382. MR**21**#4405. MR**0105667 (21:4405)****[GJ]**. L. Gillman and M. Jerison,*Rings of continuous functions*, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR**22**#6994. MR**0116199 (22:6994)****[H]**. E. Hewitt,*On two problems of Urysohn*, Ann. of Math. (2)**47**(1946), 503-509. MR**8**, 165. MR**0017527 (8:165g)****[K]**. D. Kullman,*A note on developable spaces and -spaces*(to appear).**[M]**. E. Marczewski,*Séparabilité et multiplication cartésienne des espaces topologiques*, Fund. Math.**34**(1947), 127-143. MR**9**, 98. MR**0021680 (9:98b)****[M]**. R. H. Marty,*Mazur theorem and -adic spaces*, Doctoral Dissertation, Pennsylvania State University, University Park, Pa., 1969.**[M]**. S. Mazur,*On continuous mappings on Cartesian products*, Fund. Math.**39**(1953), 229-238. MR**14**, 1107. MR**0055663 (14:1107d)****[M]**. E. Michael,*A note on intersections*, Proc. Amer. Math. Soc.**13**(1962), 281-283. MR**24**#A3070. MR**0133236 (24:A3070)****[M]**. A. Miščenko,*Several theorems on products of topological spaces*, Fund. Math.**58**(1966), 259-284. (Russian) MR**33**#4884a. MR**0196697 (33:4884a)****[RS]**. K. A. Ross and A. H. Stone,*Products of separable spaces*, Amer. Math. Monthly**71**(1964), 398-403. MR**29**#1611. MR**0164314 (29:1611)****[S]**. N. A. Šanin,*A theorem from the general theory of sets*, C. R. (Dokl.) Acad. Sci. URSS**53**(1946), 399-400. MR**8**, 333. MR**0018814 (8:333i)****[S]**. -,*On intersection of open subsets in the product of topological spaces*, C. R. (Dokl.) Acad. Sci. URSS**53**(1946), 499-501. MR**8**, 334. MR**0018815 (8:334a)****[S]**. -,*On the product of topological spaces*, Trudy Mat. Inst. Steklov.**24**(1948), 112 pp. (Russian) MR**10**, 287. MR**0027310 (10:287b)****[U]**. M. Ulmer,*-embedded II-spaces*, Notices Amer. Math. Soc.**16**(1969), 849. Abstract #69T-G105.**[U]**. -,*Continuous functions on product spaces*, Doctoral Dissertation, Wesleyan University, Middletown, Conn., 1970.**[U]**. -,*Functions on product spaces*, Notices Amer. Math. Soc.**16**(1969), 986-987. Abstract #69T-G134.**[U]**. -,*The countable chain condition*, Notices Amer. Math. Soc.**17**(1970), 462-463. Abstract #70T-G24.**[V]**. G. Vidossich,*Two remarks on A. Gleason's factorization theorem*, Bull. Amer. Math. Soc.**76**(1970), 370-371. MR**41**#1021. MR**0256365 (41:1021)****[Y]**. J. N. Younglove,*A locally connected, complete Moore space on which every real-valued continuous function is constant*, Proc. Amer. Math. Soc.**20**(1969), 527-530. MR**40**#1992. MR**0248741 (40:1992)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54.25

Retrieve articles in all journals with MSC: 54.25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0288721-2

Keywords:
Functions depending on countably many coordinates,
pseudo- -compact spaces,
countable chain condition,
realcompactifications of infinite products,
infinite products

Article copyright:
© Copyright 1972
American Mathematical Society