| Title:
|
Axiom $T_D$ and the Simmons sublocale theorem (English) |
| Author:
|
Picado, Jorge |
| Author:
|
Pultr, Aleš |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
60 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
541-551 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
More precisely, we are analyzing some of H. Simmons, S.\,B. Niefield and K.\,I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom $T_D$ for the relation of certain degrees of scatteredness but did not emphasize its role in the relation {between} sublocales and subspaces. S.\,B. Niefield and K.\,I. Rosenthal just mention this axiom in a remark about Simmons' result. In this paper we show that the role of $T_D$ in this question is crucial. Concentration on the properties of $T_D$-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem. (English) |
| Keyword:
|
frame |
| Keyword:
|
locale |
| Keyword:
|
sublocale |
| Keyword:
|
coframe of sublocales |
| Keyword:
|
spatial sublocale |
| Keyword:
|
induced sublocale |
| Keyword:
|
$T_D$-separation |
| Keyword:
|
covered prime element |
| Keyword:
|
scattered space |
| Keyword:
|
weakly scattered space |
| MSC:
|
06D22 |
| MSC:
|
54D10 |
| idZBL:
|
Zbl 07177889 |
| idMR:
|
MR4061362 |
| DOI:
|
10.14712/1213-7243.2019.030 |
| . |
| Date available:
|
2020-02-10T16:50:12Z |
| Last updated:
|
2022-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147971 |
| . |
| Reference:
|
[1] Aull C. E., Thron W. J.: Separation axioms between $T_0$ and $T_1$.Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24 (1962), 26–37. MR 0138082, 10.1016/S1385-7258(62)50003-6 |
| Reference:
|
[2] Banaschewski B., Pultr A.: Variants of openness.Appl. Categ. Structures 2 (1994), no. 4, 331–350. MR 1300720, 10.1007/BF00873038 |
| Reference:
|
[3] Banaschewski B., Pultr A.: Pointfree aspects of the $T_D$ axiom of classical topology.Quaest. Math. 33 (2010), no. 3, 369–385. MR 2755527, 10.2989/16073606.2010.507327 |
| Reference:
|
[4] Banaschewski B., Pultr A.: On covered prime elements and complete homomorphisms of frames.Quaest. Math. 37 (2014), no. 3, 451–454. MR 3285298, 10.2989/16073606.2013.780000 |
| Reference:
|
[5] Johnstone P. T.: Stone Spaces.Cambridge Studies in Advanced Mathematics, 3, Cambridge University Press, Cambridge, 1982. Zbl 0586.54001, MR 0698074 |
| Reference:
|
[6] Liu Y., Luo M.: $T_D$ property and spatial sublocales.Acta Math. Sinica (N.S.) 11 (1995), no. 3, 324–336. MR 1418002, 10.1007/BF02265398 |
| Reference:
|
[7] Niefield S. B., Rosenthal K. I.: Spatial sublocales and essential primes.Topology Appl. 26 (1987), no. 3, 263–269. MR 0904472, 10.1016/0166-8641(87)90046-0 |
| Reference:
|
[8] Picado J., Pultr A.: Frames and Locales, Topology without Points.Frontiers in Mathematics, 28, Springer, Basel, 2012. MR 2868166 |
| Reference:
|
[9] Pultr A., Tozzi A.: Separation axioms and frame representation of some topological facts.Appl. Categ. Structures 2 (1994), no. 1, 107–118. MR 1283218, 10.1007/BF00878507 |
| Reference:
|
[10] Simmons H.: The lattice theoretic part of topological separation properties.Proc. Edinburgh Math. Soc. (2) 21 (1978/79), no. 1, 41–48. MR 0493959 |
| Reference:
|
[11] Simmons H.: Spaces with Boolean assemblies.Colloq. Math. 43 (1980), no. 1, 23–29. MR 0615967, 10.4064/cm-43-1-23-39 |
| Reference:
|
[12] Thron W. J.: Lattice-equivalence of topological spaces.Duke Math. J. 29 (1962), 671–679. MR 0146787, 10.1215/S0012-7094-62-02968-X |
| . |
