Another view of metrizability

Hung, H. H.

doi:10.1090/S0002-9939-1987-0908667-X

Another view of metrizability
HTML articles powered by AMS MathViewer

by H. H. Hung

Proc. Amer. Math. Soc. 101 (1987), 551-554

DOI: https://doi.org/10.1090/S0002-9939-1987-0908667-X

PDF | Request permission

Abstract:

A fact long considered unsatisfactory about the classical metrization theorem of Alexandroff-Urysohn is that it expresses metrizability as a countable uniformity, uniformity itself being almost the former. In view of their unification, the classical theorems, with the exception of Arhangel’skiĭ’s regular open base theorem, are all really subject to the same criticism, to which our theorem here is an answer. We give a generalization here of Arhangel’skiĭ’s, of which Arhangel’skiĭ’s itself, the fundamental theorem of Alexandroff-Urysohn, A. H. Frink’s, and the Double Sequence Theorem of Nagata are all obvious special cases.

References

Une condition necessaire et suffisante pour qu’une class

soit une class

177

A. Arhangel′skiĭ, On the metrization of topological spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 589–595 (Russian, with English summary). MR 125561
A. V. Arhangel′skiĭ, Some metrization theorems, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 139–145 (Russian). MR 0156318
Dennis K. Burke, Pseudo-open mappings from topological sums, Proc. Amer. Math. Soc. 74 (1979), no. 1, 191–196. MR 521897, DOI 10.1090/S0002-9939-1979-0521897-X
J. Chaber, M. M. Čoban, and K. Nagami, On monotonic generalizations of Moore spaces, Čech complete spaces and $p$-spaces, Fund. Math. 84 (1974), no. 2, 107–119. MR 343244, DOI 10.4064/fm-84-2-107-119
P. J. Collins, G. M. Reed, A. W. Roscoe, and M. E. Rudin, A lattice of conditions on topological spaces, Proc. Amer. Math. Soc. 94 (1985), no. 3, 487–496. MR 787900, DOI 10.1090/S0002-9939-1985-0787900-X
P. J. Collins and A. W. Roscoe, Criteria for metrisability, Proc. Amer. Math. Soc. 90 (1984), no. 4, 631–640. MR 733418, DOI 10.1090/S0002-9939-1984-0733418-9
A. H. Frink, Distance functions and the metrization problem, Bull. Amer. Math. Soc. 43 (1937), no. 2, 133–142. MR 1563501, DOI 10.1090/S0002-9904-1937-06509-8
Gary Gruenhage and Phillip Zenor, Proto-metrizable spaces, Houston J. Math. 3 (1977), no. 1, 47–53. MR 442895
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
H. H. Hung, A contribution to the theory of metrization, Canadian J. Math. 29 (1977), no. 6, 1145–1151. MR 454929, DOI 10.4153/CJM-1977-113-6

On a theorem of Arhangel’skiĭ

82

H. H. Hung, A refinement on Michael’s characterization of paracompactness, Proc. Amer. Math. Soc. 83 (1981), no. 1, 179–182. MR 620008, DOI 10.1090/S0002-9939-1981-0620008-9
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
Harold W. Martin, A note on the Frink metrization theorem, Rocky Mountain J. Math. 6 (1976), no. 1, 155–157. MR 391035, DOI 10.1216/RMJ-1976-6-1-155
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
Jun-iti Nagata, A contribution to the theory of metrization, J. Inst. Polytech. Osaka City Univ. Ser. A 8 (1957), 185–192. MR 97789
Jun-iti Nagata, Modern general topology, 2nd ed., North-Holland Mathematical Library, vol. 33, North-Holland Publishing Co., Amsterdam, 1985. MR 831659
R. H. Sorgenfrey, On the topological product of paracompact spaces, Bull. Amer. Math. Soc. 53 (1947), 631–632. MR 20770, DOI 10.1090/S0002-9904-1947-08858-3
J. M. Worrell Jr. and H. H. Wicke, Characterizations of developable topological spaces, Canadian J. Math. 17 (1965), 820–830. MR 182945, DOI 10.4153/CJM-1965-080-3