Factoring functions on Cartesian products

Authors:
N. Noble and Milton Ulmer

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

MSC:
Primary 54.25

MathSciNet review:
0288721

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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]**Steve Armentrout,*A Moore space on which every real-valued continuous function is constant*, Proc. Amer. Math. Soc.**12**(1961), 106–109. MR**0120615**, 10.1090/S0002-9939-1961-0120615-3**[C]**W. W. Comfort,*A nonpseudocompact product space whose finite subproducts are pseudocompact*, Math. Ann.**170**(1967), 41–44. MR**0210070****[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**0107222****[CN]**. W. W. Comfort and S. Negrepontis,*Ultrafilters and the Stone-Čech compactification*(to appear).**[D]**Roy O. Davies,*An intersection theorem of Erdős and Rado*, Proc. Cambridge Philos. Soc.**63**(1967), 995–996. MR**0215735****[E]**R. Engelking,*On functions defined on Cartesian products*, Fund. Math.**59**(1966), 221–231. MR**0203697****[ER]**P. Erdős and R. Rado,*Intersection theorems for systems of sets*, J. London Math. Soc.**35**(1960), 85–90. MR**0111692****[F]**Zdeněk Frolík,*On two problems of W. W. Comfort*, Comment. Math. Univ. Carolinae**8**(1967), 139–144. MR**0210071****[G]**Irving Glicksberg,*Stone-Čech compactifications of products*, Trans. Amer. Math. Soc.**90**(1959), 369–382. MR**0105667**, 10.1090/S0002-9947-1959-0105667-4**[GJ]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199****[H]**Edwin Hewitt,*On two problems of Urysohn*, Ann. of Math. (2)**47**(1946), 503–509. MR**0017527****[K]**. D. Kullman,*A note on developable spaces and -spaces*(to appear).**[M]**Edward Marczewski,*Séparabilité et multiplication cartésienne des espaces topologiques*, Fund. Math.**34**(1947), 127–143 (French). MR**0021680****[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**(1952), 229–238 (1953). MR**0055663****[M]**E. Michael,*A note on intersections*, Proc. Amer. Math. Soc.**13**(1962), 281–283. MR**0133236**, 10.1090/S0002-9939-1962-0133236-4**[M]**A. Miščenko,*Several theorems on products of topological spaces*, Fund. Math.**58**(1966), 259–284 (Russian). MR**0196697****[RS]**K. A. Ross and A. H. Stone,*Products of separable spaces*, Amer. Math. Monthly**71**(1964), 398–403. MR**0164314****[S]**N. A. Shanin,*A theorem from the general theory of sets*, C. R. (Doklady) Acad. Sci. URSS (N.S.)**53**(1946), 399–400. MR**0018814****[S]**N. A. Shanin,*On intersection of open subsets in the product of topological spaces*, C. R. (Doklady) Acad. Sci. URSS (N.S.)**53**(1946), 499–501. MR**0018815****[S]**N. A. Šanin,*On the product of topological spaces*, Trudy Mat. Inst. Steklov.**24**(1948), 112 (Russian). MR**0027310****[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]**Giovanni Vidossich,*Two remarks on A. Gleason’s factorization theorem*, Bull. Amer. Math. Soc.**76**(1970), 370–371. MR**0256365**, 10.1090/S0002-9904-1970-12482-X**[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**0248741**, 10.1090/S0002-9939-1969-0248741-6

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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