Directed inverse limits of spatial locales

Authors:
Wei He and Till Plewe

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2811-2814

MSC (2000):
Primary 18B30, 54B30, 54D30, 54D45

Published electronically:
May 8, 2002

MathSciNet review:
1908261

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Abstract | References | Similar Articles | 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.

**1.**John R. Isbell,*Atomless parts of spaces*, Math. Scand.**31**(1972), 5–32. MR**0358725****2.**John Isbell,*Product spaces in locales*, Proc. Amer. Math. Soc.**81**(1981), no. 1, 116–118. MR**589150**, 10.1090/S0002-9939-1981-0589150-5**3.**John Isbell,*Direct limits of meet-continuous lattices*, J. Pure Appl. Algebra**23**(1982), no. 1, 33–35. MR**638120**, 10.1016/0022-4049(82)90076-7**4.**John Isbell,*First steps in descriptive theory of locales*, Trans. Amer. Math. Soc.**327**(1991), no. 1, 353–371. MR**1091230**, 10.1090/S0002-9947-1991-1091230-6**5.**Peter T. Johnstone,*Stone spaces*, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. MR**698074****6.**André Joyal and Myles Tierney,*An extension of the Galois theory of Grothendieck*, Mem. Amer. Math. Soc.**51**(1984), no. 309, vii+71. MR**756176**, 10.1090/memo/0309**7.**Till Plewe,*Localic products of spaces*, Proc. London Math. Soc. (3)**73**(1996), no. 3, 642–678. MR**1407464**, 10.1112/plms/s3-73.3.642**8.**Till Plewe,*Countable products of absolute 𝐶_{𝛿} spaces*, Proceedings of the International Conference on Set-theoretic Topology and its Applications (Matsuyama, 1994), 1996, pp. 39–50. MR**1425924**, 10.1016/S0166-8641(96)00042-9**9.**J. J. C. Vermeulen,*Proper maps of locales*, J. Pure Appl. Algebra**92**(1994), no. 1, 79–107. MR**1259670**, 10.1016/0022-4049(94)90047-7**10.**Wei He and Yingming Liu,*Steenrod’s theorem for locales*, Math. Proc. Cambridge Philos. Soc.**124**(1998), no. 2, 305–307. MR**1631111**, 10.1017/S0305004198002606

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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 1-1-1, Kusatsu-shi, Shiga 525, Japan

Email:
till@theory.cs.ritsumei.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-02-06196-8

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