Total paracompactness of real GOspaces
Authors:
Zoltán T. Balogh and Harold Bennett
Journal:
Proc. Amer. Math. Soc. 101 (1987), 753760
MSC:
Primary 54F05; Secondary 54D18
MathSciNet review:
911046
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A topological space is said to be totally paracompact (resp. totally metacompact) if every open base of it has a locally finite (resp. pointfinite) subcover. In this paper we characterize all totally paracompact GOspaces constructed on the real line. It turns out that in the class of GOspaces on the real line, total paracompactness and total metacompactness are equivalent. Another consequence of our characterization is that totally metacompact GOspaces on the real line are metrizable. Questions and partial results are given concerning total paracompactness in subspaces of real GOspaces.
 [BL]
H. R. Bennett and D. J. Lutzer, eds., Topology and ordered structures, Part 1, Mathematical Centre Tracts No. 142, Amsterdam, 1981.
 [BL]
H.
Bennet and D.
J. Lutzer, A note on perfect normality in generalized ordered
spaces, Topology and order structures, Part 2 (Amsterdam, 1981)
Math. Centre Tracts, vol. 169, Math. Centrum, Amsterdam, 1983,
pp. 19–22. MR 736689
(85h:54053)
 [Fa]
M.
J. Faber, Metrizability in generalized ordered spaces,
Mathematisch Centrum, Amsterdam, 1974. Mathematical Centre Tracts, No. 53.
MR
0418053 (54 #6097)
 [Fo]
R. M. Ford, Basis properties in dimension theory, Doctoral Dissertation, Auburn University, Auburn, Ala., 1963.
 [H]
R.
W. Heath, Screenability, pointwise paracompactness, and metrization
of Moore spaces, Canad. J. Math. 16 (1964),
763–770. MR 0166760
(29 #4033)
 [Le]
A. Lelek, Mathematical Problem Book, Univ. of Houston, Problem No. 99.
 [Lu]
D.
J. Lutzer, On generalized ordered spaces, Dissertationes Math.
Rozprawy Mat. 89 (1971), 32. MR 0324668
(48 #3018)
 [M]
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 0152985
(27 #2956), http://dx.doi.org/10.1090/S000299041963109313
 [OF]
J.
M. O’Farrell, The Sorgenfrey line is not totally
metacompact, Houston J. Math. 9 (1983), no. 2,
271–273. MR
703275 (84i:54039)
 [OF]
J. M. O'Farrell, Some results concerning the Hurewicz property, Fund. Math. (to appear).
 [S]
R.
H. Sorgenfrey, On the topological product of
paracompact spaces, Bull. Amer. Math. Soc.
53 (1947),
631–632. MR 0020770
(8,594f), http://dx.doi.org/10.1090/S000299041947088583
 [T]
R.
Telgársky, Total paracompactness and paracompact dispersed
spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.
16 (1968), 567–572 (English, with Loose Russian
summary). MR
0235517 (38 #3826)
 [TK]
R.
Telgársky and H.
Kok, The space of rationals is not absolutely paracompact,
Fund. Math. 73 (1971/72), no. 1, 75–78. MR 0293585
(45 #2662)
 [BL]
 H. R. Bennett and D. J. Lutzer, eds., Topology and ordered structures, Part 1, Mathematical Centre Tracts No. 142, Amsterdam, 1981.
 [BL]
 H. R. Bennett and D. J. Lutzer, eds., Topology and ordered structures, Part 2, Mathematical Centre Tracts No. 169, Amsterdam, 1983. MR 736689 (85h:54053)
 [Fa]
 M. F. Faber, Metrizability in generalized ordered spaces, Mathematical Centre Tracts No. 53, Amsterdam, 1974. MR 0418053 (54:6097)
 [Fo]
 R. M. Ford, Basis properties in dimension theory, Doctoral Dissertation, Auburn University, Auburn, Ala., 1963.
 [H]
 R. W. Heath, Screenability, pointwise paracompactness, and metrization of Moore spaces, Canad. J. Math. 16 (1964), 763770. MR 0166760 (29:4033)
 [Le]
 A. Lelek, Mathematical Problem Book, Univ. of Houston, Problem No. 99.
 [Lu]
 D. J. Lutzer, On generalized ordered spaces, Dissertationes Math. 89 (1971). MR 0324668 (48:3018)
 [M]
 E. Michael, The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc. 61 (1963), 375376. MR 0152985 (27:2956)
 [OF]
 J. M. O'Farrell, The Sorgenfrey Line is not totally metacompact, Houston J. Math 9 (1983), 271273. MR 703275 (84i:54039)
 [OF]
 J. M. O'Farrell, Some results concerning the Hurewicz property, Fund. Math. (to appear).
 [S]
 R. Sorgenfrey, On the topological product of paracompact spaces, Bull. Amer. Math. Soc. 53 (1947). MR 0020770 (8:594f)
 [T]
 R. Telgarsky, Total paracompactness and paracompact dispersed spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. 16 (1968), 567572. MR 0235517 (38:3826)
 [TK]
 R. Telgarsky and H. Kok, The space of rationals is not absolutely paracompact, Fund. Math. 73 (1971), 7578. MR 0293585 (45:2662)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
54F05,
54D18
Retrieve articles in all journals
with MSC:
54F05,
54D18
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198709110462
PII:
S 00029939(1987)09110462
Keywords:
Real GOspace,
totally paracompact,
totally metacompact,
dense and codense example,
metrizable
Article copyright:
© Copyright 1987
American Mathematical Society
