Topological spaces with a point finite base
Author:
C. E. Aull
Journal:
Proc. Amer. Math. Soc. 29 (1971), 411416
MSC:
Primary 54.50
MathSciNet review:
0281154
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Abstract: The principal results of the paper are as follows. A topological space with a point finite base has a disjoint base if it is either hereditarily collectionwise normal or hereditarily screenable. From a metrization theorem of Arhangel'skiĭ, it follows that a space with a point finite base is metrizable iff it is perfectly normal and collectionwise normal. A topological space with a point base is quasidevelopable in the sense of Bennett. Consequently a theorem of Čoban follows that for a topological space the following are equivalent: (a) is a metacompact normal Moore space, (b) is a perfectly normal space with a point finite base.
 [1]
P.
S. Aleksandrov, Some results in the theory of topological spaces
obtained within the last twentyfive years, Russian Math. Surveys
15 (1960), no. 2, 23–83. MR 0119181
(22 #9947)
 [2]
A.
V. Arhangel′skiĭ, Some metrization theorems,
Uspehi Mat. Nauk 18 (1963), no. 5 (113),
139–145 (Russian). MR 0156318
(27 #6242)
 [3]
C. E. Aull, Some base axioms involving denumerability, Proc. Conference Indian Institute of Technology (Kanpur, India, 1968), Publ. House Czechoslovak Acad. Sci. (to appear).
 [4]
H. R. Bennett, Quasidevelopable spaces, Topology Conference (Arizona State University, Tempe, Ariz. 1967), Arizona State University, Tempe, Ariz., 1968, pp. 314317.
 [5]
R.
H. Bing, Metrization of topological spaces, Canadian J. Math.
3 (1951), 175–186. MR 0043449
(13,264f)
 [6]
Dennis
K. Burke, On subparacompact spaces, Proc. Amer. Math. Soc. 23 (1969), 655–663. MR 0250269
(40 #3508), http://dx.doi.org/10.1090/S00029939196902502694
 [7]
M.
M. Čoban, 𝜎paracompact spaces, Vestnik Moskov.
Univ. Ser. I Mat. Meh. 24 (1969), no. 1, 20–27
(Russian, with English summary). MR 0257978
(41 #2627)
 [8]
H.
H. Corson and E.
Michael, Metrizability of certain countable unions, Illinois
J. Math. 8 (1964), 351–360. MR 0170324
(30 #562)
 [9]
R. W. Heath, On certain first countable spaces, Topology Seminar (Wisconsin, 1965), Princeton Univ. Press, Princeton, N. J., 1966, pp. 103113.
 [10]
R.
W. Heath, Screenability, pointwise paracompactness, and metrization
of Moore spaces, Canad. J. Math. 16 (1964),
763–770. MR 0166760
(29 #4033)
 [11]
F.
Burton Jones, Metrization, Amer. Math. Monthly
73 (1966), 571–576. MR 0199840
(33 #7980)
 [12]
Ernest
Michael, Pointfinite and locally finite coverings, Canad. J.
Math. 7 (1955), 275–279. MR 0070147
(16,1138c)
 [13]
A. Miščenko, Spaces with pointcountable base, Dokl. Akad. Nauk SSSR 144 (1962), 985988= Soviet Math. Dokl. 3 (1962), 855858. MR 25 #1537.
 [14]
Keiô
Nagami, Paracompactness and strong screenability, Nagoya Math.
J. 8 (1955), 83–88. MR 0070148
(16,1138d)
 [15]
Lester
J. Norman, A sufficient condition for quasimetrizability of a
topological space., Portugal. Math. 26 (1967),
207–211. MR 0248740
(40 #1991)
 [16]
M.
Sion and G.
Zelmer, On quasimetrizability, Canad. J. Math.
19 (1967), 1243–1249. MR 0221470
(36 #4522)
 [17]
F. D. Tall, Settheoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1969.
 [1]
 P. S. Aleksandrov, Some results in the theory of topological spaces obtained within the last twentyfive years, Uspehi Mat. Nauk 15 (1960), no. 2 (92), 2595 = Russian Math. Surveys 15 (1960), no. 2, 2383. MR 22 #9947. MR 0119181 (22:9947)
 [2]
 A. Arhangel'skiĭ, Some metrization theorems, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 139145. (Russian) MR 27 #6242. MR 0156318 (27:6242)
 [3]
 C. E. Aull, Some base axioms involving denumerability, Proc. Conference Indian Institute of Technology (Kanpur, India, 1968), Publ. House Czechoslovak Acad. Sci. (to appear).
 [4]
 H. R. Bennett, Quasidevelopable spaces, Topology Conference (Arizona State University, Tempe, Ariz. 1967), Arizona State University, Tempe, Ariz., 1968, pp. 314317.
 [5]
 R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175186. MR 13, 264. MR 0043449 (13:264f)
 [6]
 D. K. Burke, On subparacompact spaces, Proc. Amer. Math. Soc. 23 (1969), 655663. MR 40 #3508. MR 0250269 (40:3508)
 [7]
 M. M. Čoban, On paracompact spaces, Vestnik Moskov. Univ. Ser. I Math. Meh. 1969, 2027. MR 0257978 (41:2627)
 [8]
 H. Corson and E. Michael, Metrizability of certain countable unions, Illinois J. Math. 8 (1964), 351360. MR 30 #562. MR 0170324 (30:562)
 [9]
 R. W. Heath, On certain first countable spaces, Topology Seminar (Wisconsin, 1965), Princeton Univ. Press, Princeton, N. J., 1966, pp. 103113.
 [10]
 , Screenability, pointwise paracompactness, and metrization of Moore spaces, Canad. J. Math. 16 (1964), 763770. MR 29 #4033. MR 0166760 (29:4033)
 [11]
 F. B. Jones, Metrization, Amer. Math. Monthly 73 (1966), 571576. MR 33 #7980. MR 0199840 (33:7980)
 [12]
 E. A. Michael, Pointfinite and locally finite coverings, Canad. J. Math. 7 (1955), 275279. MR 16, 1138. MR 0070147 (16:1138c)
 [13]
 A. Miščenko, Spaces with pointcountable base, Dokl. Akad. Nauk SSSR 144 (1962), 985988= Soviet Math. Dokl. 3 (1962), 855858. MR 25 #1537.
 [14]
 K. Nagami, Paracompactness and strong screenability, Nagoya Math. J. 8 (1955), 8388. MR 16, 1138. MR 0070148 (16:1138d)
 [15]
 L. J. Norman, A sufficient condition for quasimetrizability of a topological space, Portugal. Math. 26 (1967), 207211. MR 40 #1991. MR 0248740 (40:1991)
 [16]
 M. Sion and G. Zelmer, On quasimetrizability, Canad. J. Math. 19 (1967), 12431249. MR 36 #4522. MR 0221470 (36:4522)
 [17]
 F. D. Tall, Settheoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1969.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197102811549
PII:
S 00029939(1971)02811549
Keywords:
point finite base,
disjoint base,
quasidevelopment,
Moore space,
hereditarily collectionwise normal,
hereditarily screenable,
metacompact or pointwise paracompact,
subparacompact or screenable
Article copyright:
© Copyright 1971
American Mathematical Society
