| Title:
|
Order intervals in $C(K)$. Compactness, coincidence of topologies, metrizability (English) |
| Author:
|
Lipecki, Zbigniew |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
295-306 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $K$ be a compact space and let $C(K)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C(K)$, including the following ones: (i) $\{x>0\}$ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. \noindent Moreover, the weak topology and that of pointwise convergence coincide on $[0,x]$ if and only if $\{x>0\}$ is scattered. Finally, the weak topology on $[0,x]$ is metrizable if and only if the topology of pointwise convergence on $[0,x]$ is such if and only if $\{x>0\}$ is countable. (English) |
| Keyword:
|
real linear lattice |
| Keyword:
|
order interval |
| Keyword:
|
locally solid |
| Keyword:
|
Banach lattice $C(K)$ |
| Keyword:
|
strongly compact |
| Keyword:
|
weakly compact |
| Keyword:
|
pointwise compact |
| Keyword:
|
coincidence of topologies |
| Keyword:
|
metrizable |
| Keyword:
|
scattered |
| Keyword:
|
Čech--Stone compactification |
| MSC:
|
46A40 |
| MSC:
|
46B42 |
| MSC:
|
46E05 |
| MSC:
|
54C35 |
| MSC:
|
54D30 |
| idZBL:
|
Zbl 07655801 |
| idMR:
|
MR4542790 |
| DOI:
|
10.14712/1213-7243.2022.006 |
| . |
| Date available:
|
2023-02-01T12:03:11Z |
| Last updated:
|
2024-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151477 |
| . |
| Reference:
|
[1] Aliprantis C. D., Burkinshaw O.: Positive Operators.Pure and Applied Mathematics, 119, Academic Press, Orlando, 1985. Zbl 1098.47001, MR 0809372 |
| Reference:
|
[2] Aliprantis C. D., Burkinshaw O.: Locally Solid Riesz Spaces with Applications to Economics.Mathematical Surveys and Monographs, 105, American Mathematical Society, Providence, 2003. MR 2011364, 10.1090/surv/105 |
| Reference:
|
[3] Arkhangel'skiĭ A. V.: Topological Function Spaces.Mathematics and Its Applications (Soviet Series), 78, Kluwer Academic Publishers Group, Dordrecht, 1992. MR 1144519, 10.1007/978-94-011-2598-7_4 |
| Reference:
|
[4] Engelking R.: General Topology.Monografie Matematyczne, 60, PWN—Polish Scientific Publishers, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[5] Floret K.: Weakly Compact Sets.Lectures held at S.U.N.Y., Buffalo, 1978, Lecture Notes in Mathematics, 801, Springer, Berlin, 1980. MR 0576235 |
| Reference:
|
[6] Lipecki Z.: Order intervals in Banach lattices and their extreme points.Colloq. Math. 160 (2020), no. 1, 119–132. MR 4071818, 10.4064/cm7726-5-2019 |
| Reference:
|
[7] Lipecki Z.: Compactness of order intervals in a locally solid linear lattice.Colloq. Math. 168 (2022), no. 2, 297–309. MR 4416011, 10.4064/cm8624-11-2021 |
| Reference:
|
[8] van Mill J.: The Infinite-Dimensional Topology of Function Spaces.North-Holland Mathematical Library, 64, North-Holland Publishing Co., Amsterdam, 2001. Zbl 0969.54003, MR 1851014 |
| Reference:
|
[9] Semadeni Z.: Banach Spaces of Continuous Functions. Vol. I.Monografie Matematyczne, 55, PWN—Polish Scientific Publishers, Warszawa, 1971. MR 0296671 |
| . |
