Directed inverse limits of spatial locales
Authors:
Wei He and Till Plewe
Journal:
Proc. Amer. Math. Soc. 130 (2002), 28112814
MSC (2000):
Primary 18B30, 54B30, 54D30, 54D45
Published electronically:
May 8, 2002
MathSciNet review:
1908261
Fulltext PDF Free Access
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Additional Information
Abstract: In this note we consider spatiality of directed inverse limits of spatial locales. We give an example which shows that directed inverse limits of compact spatial locales are not necessarily spatial. This answers a question posed by John Isbell. We also give a condition which, if satisfied by the maps of a directed inverse system, implies that taking limits preserves local compactness and hence produces spatial locales.
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J. Isbell, Product spaces in locales, Proc. Amer. Math. Soc. 81 (1981), 116118. MR 82c:54006
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J. Isbell, Direct limits of meetcontinuous lattices, J. Pure App. Algebra 23 (1982), 3335. MR 83a:18015
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T. Plewe, Localic products of spaces, Proc. London Math. Soc. 73 (1996), 642678. MR 97d:54012
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T. Plewe, Countable products of absolute Cdelta spaces, Topology Appl. 74 (1996), 3950. MR 97i:54015
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J. J. C. Vermeulen, Proper maps of locales, J. Pure App. Algebra 92 (1994), 79107. MR 95d:54009
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He Wei and Liu Ying Ming, Steenrod's theorem for locales, Math. Proc. Cambridge Philos. Soc. (1998), 124, 305307. MR 99c:18001
 1.
 J. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972), 532. MR 50:11184
 2.
 J. Isbell, Product spaces in locales, Proc. Amer. Math. Soc. 81 (1981), 116118. MR 82c:54006
 3.
 J. Isbell, Direct limits of meetcontinuous lattices, J. Pure App. Algebra 23 (1982), 3335. MR 83a:18015
 4.
 J. Isbell, First steps in descriptive theory of locales, Trans. Amer. Math. Soc. 327 (1991), 353371. MR 92b:54078
 5.
 P.T. Johnstone, Stone Spaces, Cambridge University Press 1982. MR 85f:54002
 6.
 A. Joyal and M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. No. 309, vol. 51, 1984. MR 86d:18002
 7.
 T. Plewe, Localic products of spaces, Proc. London Math. Soc. 73 (1996), 642678. MR 97d:54012
 8.
 T. Plewe, Countable products of absolute Cdelta spaces, Topology Appl. 74 (1996), 3950. MR 97i:54015
 9.
 J. J. C. Vermeulen, Proper maps of locales, J. Pure App. Algebra 92 (1994), 79107. MR 95d:54009
 10.
 He Wei and Liu Ying Ming, Steenrod's theorem for locales, Math. Proc. Cambridge Philos. Soc. (1998), 124, 305307. MR 99c:18001
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Additional Information
Wei He
Affiliation:
Department of Mathematics, Shaan Xi Normal University, Xi’an 710062, People’s Republic of China
Address at time of publication:
Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
Email:
weihe@snnu.edu.cn, weihe@njnu.edu.cn
Till Plewe
Affiliation:
Department of Science and Engineering, Ritsumeikan University, Noji Higashi 111, Kusatsushi, Shiga 525, Japan
Email:
till@theory.cs.ritsumei.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993902061968
PII:
S 00029939(02)061968
Keywords:
Directed inverse limits,
spatial locales,
locally compact spaces,
locally compact locales,
compact locales
Received by editor(s):
May 17, 1998
Received by editor(s) in revised form:
October 30, 2000
Published electronically:
May 8, 2002
Additional Notes:
The first author was supported by a grant of the NSF of China
Communicated by:
Alan Dow
Article copyright:
© Copyright 2002 American Mathematical Society
