Article
| 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 |
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| Category: | math |
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| 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 |
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| Date available: | 2023-02-01T12:03:11Z |
| Last updated: | 2023-04-20 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151477 |
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| 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 |
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