| Title:
|
On open maps and related functions over the Salbany compactification (English) |
| Author:
|
Nxumalo, Mbekezeli |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
60 |
| Issue:
|
1 |
| Year:
|
2024 |
| Pages:
|
21-33 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Given a topological space $X$, let $\mathcal{U}X$ and $\eta _{X}\colon X\rightarrow \mathcal{U}X$ denote, respectively, the Salbany compactification of $X$ and the compactification map called the Salbany map of $X$. For every continuous function $f\colon X\rightarrow Y$, there is a continuous function $\mathcal{U}f\colon \mathcal{U}X\rightarrow \mathcal{U}Y$, called the Salbany lift of $f$, satisfying $(\mathcal{U}f)\circ \eta _{X}=\eta _{Y}\circ f$. If a continuous function $f\colon X\rightarrow Y$ has a stably compact codomain $Y$, then there is a Salbany extension $F\colon \mathcal{U}X\rightarrow Y$ of $f$, not necessarily unique, such that $F\circ \eta _{X}=f$. In this paper, we give a condition on a space such that its Salbany map is open. In particular, we prove that in a class of Hausdorff spaces, the spaces with open Salbany maps are precisely those that are almost discrete. We also investigate openness of the Salbany lift and a Salbany extension of a continuous function. Related to open continuous functions are initial maps as well as nearly open maps. It turns out that the Salbany map of every space is both initial and nearly open. We repeat the procedure done for openness of Salbany maps, Salbany lifts and Salbany extensions to their initiality and nearly openness. (English) |
| Keyword:
|
ultrafilter |
| Keyword:
|
ultrafilter space |
| Keyword:
|
compact space |
| Keyword:
|
compactification |
| Keyword:
|
open map |
| Keyword:
|
initial map |
| Keyword:
|
nearly open map |
| Keyword:
|
compact-open basis |
| Keyword:
|
spectral space |
| Keyword:
|
quasi-spectral space |
| MSC:
|
54D35 |
| MSC:
|
54D80 |
| DOI:
|
10.5817/AM2024-1-21 |
| . |
| Date available:
|
2024-02-07T14:11:57Z |
| Last updated:
|
2024-02-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152025 |
| . |
| Reference:
|
[1] Adjei, I., Dube, T.: The Banaschewski extension and some variants of openness.Houston J. Math. 47 (2021), 245–261. MR 4447972 |
| Reference:
|
[2] Al-Hajri, M., Belaid, K., Echi, O.: Stone spaces and compactifications.Pure Math. Sci. 2 (2) (2013), 75–81. 10.12988/pms.2013.13010 |
| Reference:
|
[3] Bentley, H.L., Herrlich, H.: A characterization of the prime closed filter compactification of a topological space.Quaest. Math. 25 (3) (2002), 381–396. MR 1931288, 10.2989/16073600209486024 |
| Reference:
|
[4] Bezhanishvili, G., Gabelaia, D., Jibladze, M., Morandi, P.J.: Profinite topological spaces.Theory Appl. Categ. 30 (53) (2015), 1841–1863. MR 3595566 |
| Reference:
|
[5] Bezhanishvili, G., Mines, R., Morandi, P.J.: Topo-canonical completions of closure algebras and Heyting algebras.Algebra Universalis 58 (2008), 1–34. MR 2375279, 10.1007/s00012-007-2032-2 |
| Reference:
|
[6] D., Harris: Closed images of the Wallman compactification.Proc. Amer. Math. Soc. 42 (1) (1974), 312–319. MR 0343238, 10.1090/S0002-9939-1974-0343238-9 |
| Reference:
|
[7] Dimov, G.: Open and other kinds of map extensions over zero-dimensional local compactifications.Topology Appl. 157 (14) (2010), 2251–2260. MR 2670501 |
| Reference:
|
[8] Engelking, R.: General topology.Sigma Ser. Pure Math., vol. 6, Heldermann Verlag, Berlin, 1989, pp. viii+529 pp. MR 1039321 |
| Reference:
|
[9] Kuratowski, K.: Topology.vol. 1, Polish Scientific Publishers, Academic Press, Warsaw, New York and London, 1966. MR 0217751 |
| Reference:
|
[10] Nxumalo, M.S.: Ultrafilters and compactification.Master's thesis, University of the Western Cape, 2019, 72 pp. |
| Reference:
|
[11] Salbany, S.: Ultrafilter spaces and compactifications.Port. Math. 57 (4) (2000), 481–492. MR 1807357 |
| Reference:
|
[12] Smyth, M.B.: Stable compactification 1.J. Lond. Math. Soc. 2 (2) (1992), 321–340. MR 1171559, 10.1112/jlms/s2-45.2.321 |
| . |
