Criteria for metrisability
HTML articles powered by AMS MathViewer
 by P. J. Collins and A. W. Roscoe PDF
 Proc. Amer. Math. Soc. 90 (1984), 631640 Request permission
Abstract:
A simple condition on the local bases of a first countable space is shown to imply metrisability, and some new and some wellknown metrisation theorems are deduced. Weakening the condition gives new classes of spaces distinct from the class of metrisable spaces.References

P. S. Alexandroff and P. Urysohn, Une condition nécessaire et suffisante pour qu’une classe (L) soit une classe (B), C. R. Acad. Sci. Paris 177 (1923), 12741276.
A. Arhangel’skii, New criteria for the paracompaetness and metrizability of an arbitrary ${T_1}$space, Soviet Math. Dokl. 2 (1961), 13671369. (Russian)
 R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175–186. MR 43449, DOI 10.4153/cjm19510223 N. Bourbaki, General topology, Part 2, AddisonWesley, Reading, Mass. and Hermann, Paris, 1966. P. J. Collins and A. W. Roscoe, A note on metrisation (to appear).
 Geoffrey D. Creede, Concerning semistratifiable spaces, Pacific J. Math. 32 (1970), 47–54. MR 254799
 Ralph Fox, Solution of the $\gamma$space problem, Proc. Amer. Math. Soc. 85 (1982), no. 4, 606–608. MR 660614, DOI 10.1090/S0002993919820660614X
 A. H. Frink, Distance functions and the metrization problem, Bull. Amer. Math. Soc. 43 (1937), no. 2, 133–142. MR 1563501, DOI 10.1090/S000299041937065098
 Sitiro Hanai, On closed mappings, Proc. Japan Acad. 30 (1954), 285–288. MR 64390
 R. W. Heath, D. J. Lutzer, and P. L. Zenor, Monotonically normal spaces, Trans. Amer. Math. Soc. 178 (1973), 481–493. MR 372826, DOI 10.1090/S00029947197303728262
 R. E. Hodel, Spaces defined by sequences of open covers which guarantee that certain sequences have cluster points, Duke Math. J. 39 (1972), 253–263. MR 293580
 W. F. Lindgren and P. Fletcher, Locally quasiuniform spaces with countable bases, Duke Math. J. 41 (1974), 231–240. MR 341422
 W. F. Lindgren and P. J. Nyikos, Spaces with bases satisfying certain order and intersection properties, Pacific J. Math. 66 (1976), no. 2, 455–476. MR 445452
 Louis F. McAuley, A relation between perfect separability, completeness, and normality in semimetric spaces, Pacific J. Math. 6 (1956), 315–326. MR 80907
 E. Michael, Yet another note on paracompact spaces, Proc. Amer. Math. Soc. 10 (1959), 309–314. MR 105668, DOI 10.1090/S00029939195901056681
 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/S000299041963109313 R. L. Moore, A set of axioms for plane analysis situs, Fund. Math. 25 (1935), 1328.
 Kiiti Morita, On the simple extension of a space with respect to a uniformity. IV, Proc. Japan Acad. 27 (1951), 632–636. MR 52090
 Juniti Nagata, On a necessary and sufficient condition of metrizability, J. Inst. Polytech. Osaka City Univ. Ser. A 1 (1950), 93–100. MR 43448
 Juniti Nagata, A contribution to the theory of metrization, J. Inst. Polytech. Osaka City Univ. Ser. A 8 (1957), 185–192. MR 97789 —, Modern general topology, WoltersNoordhoff, Groningen, 1968.
 V. W. Niemytzki, On the “third axiom of metric space”, Trans. Amer. Math. Soc. 29 (1927), no. 3, 507–513. MR 1501402, DOI 10.1090/S00029947192715014022
 Prabir Roy, Failure of equivalence of dimension concepts for metric spaces, Bull. Amer. Math. Soc. 68 (1962), 609–613. MR 142102, DOI 10.1090/S000299041962108726
 Mary Ellen Rudin, A new proof that metric spaces are paracompact, Proc. Amer. Math. Soc. 20 (1969), 603. MR 236876, DOI 10.1090/S00029939196902368763
 Yu. Smirnov, A necessary and sufficient condition for metrizability of a topological space, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 197–200 (Russian). MR 0041420
 A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. 54 (1948), 977–982. MR 26802, DOI 10.1090/S000299041948091182
 A. H. Stone, Sequences of coverings, Pacific J. Math. 10 (1960), 689–691. MR 119189
 Jingoro Suzuki, On the metrization and the completion of a space with respect to a uniformity, Proc. Japan Acad. 27 (1951), 219–223. MR 48785
 James Williams, Locally uniform spaces, Trans. Amer. Math. Soc. 168 (1972), 435–469. MR 296891, DOI 10.1090/S00029947197202968915
Additional Information
 © Copyright 1984 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 90 (1984), 631640
 MSC: Primary 54E35; Secondary 54E15
 DOI: https://doi.org/10.1090/S00029939198407334189
 MathSciNet review: 733418