| Title:
|
Base-base paracompactness and subsets of the Sorgenfrey line (English) |
| Author:
|
Popvassilev, Strashimir G. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
137 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
395-401 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A topological space $X$ is called base-base paracompact (John E. Porter) if it has an open base $\mathcal B$ such that every base ${\mathcal B' \subseteq \mathcal B}$ has a locally finite subcover $\mathcal C \subseteq \mathcal B'$. It is not known if every paracompact space is base-base paracompact. We study subspaces of the Sorgenfrey line (e.g. the irrationals, a Bernstein set) as a possible counterexample. (English) |
| Keyword:
|
base-base paracompact space |
| Keyword:
|
coarse base |
| Keyword:
|
Sorgenfrey irrationals |
| Keyword:
|
totally imperfect set |
| MSC:
|
03E15 |
| MSC:
|
26A21 |
| MSC:
|
28A05 |
| MSC:
|
54D20 |
| MSC:
|
54D70 |
| MSC:
|
54F05 |
| MSC:
|
54G20 |
| MSC:
|
54H05 |
| idZBL:
|
Zbl 1274.54075 |
| idMR:
|
MR3058271 |
| DOI:
|
10.21136/MB.2012.142995 |
| . |
| Date available:
|
2012-11-10T20:26:49Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142995 |
| . |
| Reference:
|
[1] Arhangelskii, A. V.: On the metrization of topological spaces.Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 8 (1960), 589-595 Russian \MR 0125561. MR 0125561 |
| Reference:
|
[2] Balogh, Z., Bennett, H.: Total paracompactness of real GO-spaces.Proc. Amer. Math. Soc. 101 (1987), 753-760. Zbl 0627.54016, MR 0911046, 10.1090/S0002-9939-1987-0911046-2 |
| Reference:
|
[3] Caux, P. de: Yet another property of the Sorgenfrey plane.Topology Proc. 6 (1981), 31-43. MR 0650479 |
| Reference:
|
[4] Corson, H. H., McMinn, T. J., Michael, E. A., Nagata, J.: Bases and local finiteness.Notices Amer. Math. Soc. 6 (1959), 814 (abstract). |
| Reference:
|
[5] Corson, H. H.: Collections of convex sets which cover a Banach space.Fund. Math. 49 (1960/1961), 143-145. MR 0125430, 10.4064/fm-49-2-143-145 |
| Reference:
|
[6] Curtis, D. W.: Total and absolute paracompactness.Fund. Math. 77 (1973), 277-283. Zbl 0248.54022, MR 0321005, 10.4064/fm-77-3-277-283 |
| Reference:
|
[7] Douwen, Eric K. van, Pfeffer, W.: Some properties of the Sorgenfrey line and related spaces.Pacific J. Math. 81 (1979), 371-377. MR 0547605, 10.2140/pjm.1979.81.371 |
| Reference:
|
[8] Ford, R.: Basic properties in dimension theory.Dissertation, Auburn University (1963). MR 2613829 |
| Reference:
|
[9] Gerlits, J., Nagy, Zs.: Some properties of $C(X)$, I.Topol. Appl. 14 (1982), 151-161. Zbl 0503.54020, MR 0667661, 10.1016/0166-8641(82)90065-7 |
| Reference:
|
[10] Gruenhage, Gary: A survey of $D$-spaces.Contemporary Mathematics 533 (2011), 13-28. Zbl 1217.54025, MR 2777743, 10.1090/conm/533/10502 |
| Reference:
|
[11] Lelek, A.: On totally paracompact metric spaces.Proc. Amer. Math. Soc. 19 (1968), 168-170. Zbl 0153.52602, MR 0219032, 10.1090/S0002-9939-1968-0219032-3 |
| Reference:
|
[12] Lelek, A.: Some cover properties of spaces.Fund. Math. 64 (1969), 209-218. Zbl 0175.49603, MR 0242108, 10.4064/fm-64-2-209-218 |
| Reference:
|
[13] O'Farrell, J. M.: Some methods of determining total paracompactness.Diss., Auburn Univ. (1982). |
| Reference:
|
[14] O'Farrell, J. M.: The Sorgenfrey line is not totally metacompact.Houston J. Math. 9 (1983), 271-273. Zbl 0518.54020, MR 0703275 |
| Reference:
|
[15] O'Farrell, J. M.: Construction of a Hurewicz metric space whose square is not a Hurewicz space.Fund. Math. 127 (1987), 41-43. MR 0883150, 10.4064/fm-127-1-41-43 |
| Reference:
|
[16] Popvassilev, S. G.: Base-cover paracompactness.Proc. Amer. Math. Soc. 132 (2004), 3121-3130. Zbl 1062.54022, MR 2063135, 10.1090/S0002-9939-04-07457-X |
| Reference:
|
[17] Popvassilev, S. G.: Base-family paracompactness.Houston J. Math. 32 (2006), 459-469. Zbl 1148.54008, MR 2219324 |
| Reference:
|
[18] Popvassilev, S. G.: On base-cover metacompact products.Topol. Appl. 157 (2010), 2553-2554. Zbl 1200.54011, MR 2719398, 10.1016/j.topol.2010.07.035 |
| Reference:
|
[19] Popvassilev, S. G.: Base-base, base-cover and base-family paracompactness.Contributed Problems, Zoltán Balogh Memorial Topology Conference, Miami Univ., Oxford, Ohio, Nov. 15-16 (2002), 18-19. http://notch.mathstat.muohio.edu/balog\_conference/all\_prob.pdf. |
| Reference:
|
[20] Popvassilev, S. G.: Problems by S. Popvassilev.Problems in General and Set-Theoretic Topology, 2004 Spring Topology and Dynamics Conference at the University of Alabama, Birmingham. http://www.auburn.edu/ {gruengf/confprobs.pdf}. |
| Reference:
|
[21] Porter, J. E.: Generalizations of totally paracompact spaces.Diss., Auburn Univ. (2000). |
| Reference:
|
[22] Porter, J. E.: Base-paracompact spaces.Topol. Appl. 128 (2003), 145-156. Zbl 1099.54021, MR 1956610, 10.1016/S0166-8641(02)00109-8 |
| Reference:
|
[23] Porter, J. E.: Strongly base-paracompact spaces.Comment. Math. Univ. Carolin. 44 (2003), 307-314. Zbl 1099.54021, MR 2026165 |
| Reference:
|
[24] Rudin, M. E.: Martin's axiom.Jon Barwise Handbook of mathematical logic, Studies in Logic and the Found. of Math. 90 North-Holland (1977), 491-501. MR 0457132 |
| Reference:
|
[25] Sakai, M.: Menger subsets of the Sorgenfrey line.Proc. Amer. Math. Soc. 137 (2009), 3129-3138. Zbl 1182.03077, MR 2506472, 10.1090/S0002-9939-09-09887-6 |
| Reference:
|
[26] Telgárski, R., Kok, H.: The space of rationals is not absolutely paracompact.Fund. Math. 73 (1971/72), 75-78. MR 0293585, 10.4064/fm-73-1-75-78 |
| . |
