Research article Special Issues

Covering properties of $ C_{p}\left(Y|X\right) $

  • Received: 05 February 2024 Revised: 28 April 2024 Accepted: 29 April 2024 Published: 23 May 2024
  • MSC : 46A03, 54C30

  • Let $ X $ be an infinite Tychonoff space, and $ Y $ be a topological subspace of $ X $. In this paper, we study some covering properties of the subspace $ C_{p}\left(Y|X\right) $ of $ C_{p}\left(Y\right) $ consisting of those functions $ f\in C\left(Y\right) $ which admit a continuous extension to $ X $ equipped with the relative topology of $ C_{p}\left(Y\right) $. Among other results, we show that $ \left(i\right) $ $ C_{p}(Y|X) $ has a fundamental bounded resolution if and only if $ Y $ is countable; when $ X $ is realcompact and $ Y $ is closed in $ X $, we have $ \left(ii\right) $ if $ C_{p}(Y|X) $ admits a resolution of convex compact sets that swallows the local null sequences in $ C_{p}(Y|X) $, then $ Y $ is countable and discrete; $ \left(iii\right) $ if $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets, then $ Y $ is also countable and discrete, and, as a corollary, we deduce that $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets if and only if $ C_{p}(Y|X) $ is a Polish space. We also prove that $ \left(iv\right) $ for a metrizable space $ X $, $ C_{p}\left(X\right) $ is a quasi-$ \left(LB\right) $-space if and only if $ X $ is $ \sigma $-compact, and hence for a subspace $ Y $ of $ X $, the space $ C_{p}\left(Y|X\right) $ is a quasi-$ \left(LB\right) $-space. We include some examples and observations that answer natural questions raised in this paper.

    Citation: Juan C. Ferrando, Manuel López-Pellicer, Santiago Moll-López. Covering properties of $ C_{p}\left(Y|X\right) $[J]. AIMS Mathematics, 2024, 9(7): 17743-17757. doi: 10.3934/math.2024862

    Related Papers:

  • Let $ X $ be an infinite Tychonoff space, and $ Y $ be a topological subspace of $ X $. In this paper, we study some covering properties of the subspace $ C_{p}\left(Y|X\right) $ of $ C_{p}\left(Y\right) $ consisting of those functions $ f\in C\left(Y\right) $ which admit a continuous extension to $ X $ equipped with the relative topology of $ C_{p}\left(Y\right) $. Among other results, we show that $ \left(i\right) $ $ C_{p}(Y|X) $ has a fundamental bounded resolution if and only if $ Y $ is countable; when $ X $ is realcompact and $ Y $ is closed in $ X $, we have $ \left(ii\right) $ if $ C_{p}(Y|X) $ admits a resolution of convex compact sets that swallows the local null sequences in $ C_{p}(Y|X) $, then $ Y $ is countable and discrete; $ \left(iii\right) $ if $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets, then $ Y $ is also countable and discrete, and, as a corollary, we deduce that $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets if and only if $ C_{p}(Y|X) $ is a Polish space. We also prove that $ \left(iv\right) $ for a metrizable space $ X $, $ C_{p}\left(X\right) $ is a quasi-$ \left(LB\right) $-space if and only if $ X $ is $ \sigma $-compact, and hence for a subspace $ Y $ of $ X $, the space $ C_{p}\left(Y|X\right) $ is a quasi-$ \left(LB\right) $-space. We include some examples and observations that answer natural questions raised in this paper.



    加载中


    [1] R. F. Arens, A topology for spaces of transformations, Ann. Math., 47 (1946), 480–495. https://doi.org/10.2307/1969087 doi: 10.2307/1969087
    [2] A. V. Arkhangel'skiĭ, Topological function spaces, In: Mathematics and its applications, Dordrecht: Springer, 1992.
    [3] A. V. Arkhangel'skiĭ, $C_{p}$-Theory, In: Recent progress in general topology, Elsevier, 1992, 1–56.
    [4] V. V. Arkhangel'skiĭ, M. Choban, The extension property of Tychonoff spaces and generalized retracts, C. R. Acad. Bulg. Sci., 41 (1988), 5–7.
    [5] V. I. Bogachev, O. G. Smolyanov, Topological vector spaces and their applications, Cham: Springer, 2017. https://doi.org/10.1007/978-3-319-57117-1
    [6] H. Buchwalter, J. Schmets, Sur quelques propriétés de l'espace $C_{s}\left(T\right)$, J. Math. Pures Appl., 52 (1973), 337–352.
    [7] B. Cascales, On $K$-analytic locally convex spaces, Arch. Math., 49 (1987), 232–244. https://doi.org/10.1007/BF01271663 doi: 10.1007/BF01271663
    [8] J. P. R. Christensen, Topology and Borel structure: Descriptive topology and set theory with applications to functional analysis and measure theory, New York, North-Holland Publishing Company, 1974.
    [9] R. Engelking, General topology, Heldermann Verlag, 1989.
    [10] J. C. Ferrando, Some characterizations for $\upsilon X$ to be Lindelöf $\Sigma$ of $K$-analytic in terms of $C_{p}\left(X\right)$, Topol. Appl., 156 (2009), 823–830. https://doi.org/10.1016/j.topol.2008.10.016 doi: 10.1016/j.topol.2008.10.016
    [11] J. C. Ferrando, A characterization of the existence of a fundamental bounded resolution for the space $C_{c}\left(X\right)$ in terms of $X$, J. Funct. Space., 2018 (2018), 8219246. https://doi.org/10.1155/2018/8219246 doi: 10.1155/2018/8219246
    [12] J. C. Ferrando, Descriptive topology for analysts, RACSAM, 114 (2020), 107. https://doi.org/10.1007/s13398-020-00837-z doi: 10.1007/s13398-020-00837-z
    [13] J. C. Ferrando, Existence of nice resolutions in $C_{p}\left(X\right)$ and its bidual often implies metrizability of $C_{p}\left(X\right)$, Topol. Appl., 282 (2020), 107322. https://doi.org/10.1016/j.topol.2020.107322 doi: 10.1016/j.topol.2020.107322
    [14] J. C. Ferrando, M. López-Pellicer, Covering properties of $C_{p}\left(X\right)$ and $C_{k}\left(X\right)$, Filomat, 34 (2020), 3575–3599. https://doi.org/10.2298/FIL2011575F doi: 10.2298/FIL2011575F
    [15] J. C. Ferrando, S. Gabriyelyan, J. Ka̧kol, Bounded sets structure of $C_{p}\left(X\right)$ and quasi-$(DF)$-spaces, Math. Nachr., 292 (2019), 2602–2618. https://doi.org/10.1002/mana.201800085 doi: 10.1002/mana.201800085
    [16] J. C. Ferrando, S. Gabriyelyan, J. Ka̧kol, Functional Characterizations of Countable Tychonoff Spaces, J. Convex Anal., 26 (2019), 753–760.
    [17] J. C. Ferrando, J. Ka̧kol, S. A. Saxon, Characterizing $P$-spaces in terms of $C_{p}\left(X\right)$, J. Convex Anal., 22 (2015), 905–915.
    [18] J. C. Ferrando, S. A. Saxon, The even large subspace $C_{p}\left(Y|X\right)$: Distinguished, montel, covered nicely?, submitted for publication.
    [19] L. Gillman, M. Jerison, Rings of continuous functions, 1960.
    [20] S. Gabriyelyan, Local completeness of $C_{k}\left(X\right)$, RACSAM, 117 (2023), Paper 152. https://doi.org/10.1007/s13398-023-01487-7 doi: 10.1007/s13398-023-01487-7
    [21] S. Gabriyelyan, J. Ka̧kol, Free locally convex spaces with a small base, RACSAM, 111 (2017), 575–585. https://doi.org/10.1007/s13398-016-0315-1 doi: 10.1007/s13398-016-0315-1
    [22] T. E. Gilsdorf, Valdivia's lifting theorem for non-metrizable spaces, Topol. Appl., 317 (2022), 108160. https://doi.org/10.1016/j.topol.2022.108160 doi: 10.1016/j.topol.2022.108160
    [23] H. Jarchow, Locally convex spaces, 1981.
    [24] J. Ka̧kol, W. Kubiś, M. López-Pellicer, Descriptive topology in selected topics of functional analysis, New York: Springer, 2011. https://doi.org/10.1007/978-1-4614-0529-0
    [25] G. Köthe, Topological vector spaces I, Berlin: Springer, 1983. https://doi.org/10.1007/978-3-642-64988-2
    [26] G. Köthe, Topological vector spaces II, New York: Springer, 1979. https://doi.org/10.1007/978-1-4684-9409-9
    [27] D. Lutzer, R. A. McCoy, Category in function spaces I, Pacific J. Math., 90 (1980), 145–168.
    [28] J. Orihuela, Pointwise compactness in spaces of continuous functions, J. London Math. Soc., 2 (1987), 143–152.
    [29] S. A. Saxon, Two characterizations of linear Baire spaces, Proc. Amer. Math. Soc., 45 (1974), 204–208.
    [30] V. V. Tkachuk, A space $C_{p}\left(X\right)$ is dominated by irrationals if and only if it is $K$-analytic, Acta Math. Hung., 107 (2005), 253–265. https://doi.org/10.1007/s10474-005-0194-y doi: 10.1007/s10474-005-0194-y
    [31] V. V. Tkachuk, D. B. Shakhmatov, When is space $C_{p}\left(X\right)$ $\sigma$-countably compact?, Vestnik Moskov. Univ. Ser. 1 Math. Mekh., 1 (1986), 70–72.
    [32] M. Valdivia, Topics in locally convex spaces, Elsevier, 1982.
    [33] M. Valdivia, Quasi-$LB$-spaces, J. London Math. Soc., 35 (1987), 149–168. https://doi.org/10.1112/jlms/s2-35.1.149
    [34] A. Wilansky, Topology for analysis, 1970.
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(198) PDF downloads(20) Cited by(0)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog