Products with closed projections. II
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- by N. Noble PDF
- Trans. Amer. Math. Soc. 160 (1971), 169-183 Request permission
Abstract:
Conditions under which some or all of the projections on a product space will be closed or $z$-closed are studied, with emphasis on infinite products. These results are applied to characterize normal products up to countably many factors, to characterize closed product maps up to finitely many factors, and to give conditions under which products will be countably compact, Lindelöf, paracompact, $\mathfrak {m} - \mathfrak {n}$-compact, etc. Generalizations of these results to $\mathfrak {n}$-products and box products are also given. Our easily stated results include: All powers of a ${T_1}$ space $X$ are normal if and only if $X$ is compact Hausdorff, all powers of a nontrivial closed map $p$ are closed if and only if $p$ is proper, the product of countably many Lindelöf $P$-spaces is Lindelöf; and the product of countably many countably compact sequential spaces is countably compact sequential.References
- Carlos J. R. Borges, On a counter-example of A. H. Stone, Quart. J. Math. Oxford Ser. (2) 20 (1969), 91–95. MR 253274, DOI 10.1093/qmath/20.1.91
- W. W. Comfort, A nonpseudocompact product space whose finite subproducts are pseudocompact, Math. Ann. 170 (1967), 41–44. MR 210070, DOI 10.1007/BF01362285
- W. W. Comfort and Anthony W. Hager, The projection mapping and other continuous functions on a product space, Math. Scand. 28 (1971), 77–90. MR 315657, DOI 10.7146/math.scand.a-11007
- W. W. Comfort and Anthony W. Hager, Estimates for the number of real-valued continuous functions, Trans. Amer. Math. Soc. 150 (1970), 619–631. MR 263016, DOI 10.1090/S0002-9947-1970-0263016-X I. Fleischer and S. P. Franklin, On compactness and projections, Internationale Spezialtagung für Erweiterungstheorie topologischer Strukturen und deren Anwendungen, Berlin, 1967.
- Zdeněk Frolík, The topological product of countably compact spaces, Czechoslovak Math. J. 10(85) (1960), 329–338 (English, with Russian summary). MR 117705, DOI 10.21136/CMJ.1960.100417
- Zdeněk Frolík, On two problems of W. W. Comfort, Comment. Math. Univ. Carolinae 8 (1967), 139–144. MR 210071
- Zdeněk Frolík, Sums of ultrafilters, Bull. Amer. Math. Soc. 73 (1967), 87–91. MR 203676, DOI 10.1090/S0002-9904-1967-11653-7
- Irving Glicksberg, Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369–382. MR 105667, DOI 10.1090/S0002-9947-1959-0105667-4
- G. G. Gould, Locally unbounded topological fields and box topologies on products of vector spaces, J. London Math. Soc. 36 (1961), 273–281. MR 130553, DOI 10.1112/jlms/s1-36.1.273 S. L. Gulden, On $m$-sequential spaces, Notices Amer. Math. Soc. 16 (1969), 293. Abstract #663-706.
- Anthony W. Hager, Projections of zero-sets (and the fine uniformity on a product), Trans. Amer. Math. Soc. 140 (1969), 87–94. MR 242114, DOI 10.1090/S0002-9947-1969-0242114-2
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
- C. J. Knight, Box topologies, Quart. J. Math. Oxford Ser. (2) 15 (1964), 41–54. MR 160184, DOI 10.1093/qmath/15.1.41
- E. Michael, The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc. 69 (1963), 375–376. MR 152985, DOI 10.1090/S0002-9904-1963-10931-3
- Ernest Michael, Local compactness and Cartesian products of quotient maps and $k$-spaces, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 2, 281–286 vii (1969) (English, with French summary). MR 244943, DOI 10.5802/aif.300
- K. Morita, Paracompactness and product spaces, Fund. Math. 50 (1961/62), 223–236. MR 132525, DOI 10.4064/fm-50-3-223-236
- S. Mrówka, On function spaces, Fund. Math. 45 (1958), 273–282. MR 98312, DOI 10.4064/fm-45-1-283-291
- N. Noble, Products with closed projections, Trans. Amer. Math. Soc. 140 (1969), 381–391. MR 250261, DOI 10.1090/S0002-9947-1969-0250261-4
- N. Noble, Ascoli theorems and the exponential map, Trans. Amer. Math. Soc. 143 (1969), 393–411. MR 248727, DOI 10.1090/S0002-9947-1969-0248727-6
- N. Noble, A note on $z$-closed projections, Proc. Amer. Math. Soc. 23 (1969), 73–76. MR 246271, DOI 10.1090/S0002-9939-1969-0246271-9 N. Noble, Products of quotient maps and spaces with weak topologies (to appear).
- Norman Noble, Countably compact and pseudo-compact products, Czechoslovak Math. J. 19(94) (1969), 390–397. MR 248717, DOI 10.21136/CMJ.1969.100911
- Norman Noble, A generalization of a theorem of A. H. Stone, Arch. Math. (Basel) 18 (1967), 394–395. MR 222833, DOI 10.1007/BF01898831 —, A nice proof of Tychonoff’s theorem (to appear).
- C. T. Scarborough, Closed graphs and closed projections, Proc. Amer. Math. Soc. 20 (1969), 465–470. MR 250275, DOI 10.1090/S0002-9939-1969-0250275-X
- C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131–147. MR 203679, DOI 10.1090/S0002-9947-1966-0203679-7
- R. M. Stephenson Jr., Pseudocompact spaces, Trans. Amer. Math. Soc. 134 (1968), 437–448. MR 232349, DOI 10.1090/S0002-9947-1968-0232349-6
- R. M. Stephenson Jr., Product spaces for which the Stone-Weierstrass theorem holds, Proc. Amer. Math. Soc. 21 (1969), 284–288. MR 250260, DOI 10.1090/S0002-9939-1969-0250260-8
- Hisahiro Tamano, A note on the pseudo-compactness of the product of two spaces, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 225–230. MR 120619, DOI 10.1215/kjm/1250775908
- John W. Tukey, Convergence and Uniformity in Topology, Annals of Mathematics Studies, No. 2, Princeton University Press, Princeton, N. J., 1940. MR 0002515
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 160 (1971), 169-183
- MSC: Primary 54.25
- DOI: https://doi.org/10.1090/S0002-9947-1971-0283749-X
- MathSciNet review: 0283749