Closed ideals in the functionally countable subalgebra of C(X)

Amir Veisi

doi:10.4995/agt.2022.15844

Authors

Amir Veisi Yasouj University

DOI:

https://doi.org/10.4995/agt.2022.15844

Keywords:

zero-dimensional space, functionally countable subalgebra, m-topology, closed ideal, ec-filter, ec-ideal, P-space

Abstract

In this paper, closed ideals in C_c(X), the functionally countable subalgebra of C(X), with the m_c-topology, is studied. We show that if
X is CUC-space, then C^*_c(X) with the uniform norm-topology is a Banach algebra. Closed ideals in C_c(X) as a modified countable analogue of closed ideals in C(X) with the m-topology are characterized. For a zero-dimensional space X, we show that a proper ideal in C_c(X) is closed if and only if it is an intersection of maximal ideals of C_c(X). It is also shown that every ideal in C_c(X) with the m_c-topology is closed if and only if X is a P-space if and only if every ideal in C(X) with the m-topology is closed. Moreover, for a strongly zero-dimensional space X, it is proved that a properly closed ideal in C^*_c(X) is an intersection of maximal ideals of C^*_c(X) if and only if X is pseudo compact. Finally, we show that if X is a P-space, then the family of e_c-ultrafilters and z_c-ultrafilter coincide.

Downloads

Download data is not yet available.

Author Biography

Amir Veisi, Yasouj University

Faculty of Petroleum and Gas

References

F. Azarpanah, O. A. S. Karamzadeh, Z. Keshtkar and A. R. Olfati, On maximal ideals of $C_c(X)$ and uniformity its localizations, Rocky Mountain Journal of Mathematics 48, no. 2 (2018), 345-382. https://doi.org/10.1216/RMJ-2018-48-2-345

R. Engelking, General topology, Sigma Ser. Pure Math., Vol. 6, Heldermann Verlag, Berlin, 1989.

M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova 129 (2013), 47-69. https://doi.org/10.4171/RSMUP/129-4

M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, C(X) versus its functionally countable subalgebra, The Bulletin of the Iranian Mathematical Society 45, no. 1 (2019), 173-187. https://doi.org/10.1007/s41980-018-0124-8

L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, Berlin/Heidelberg/New York, 1976.

J. Gómez-Pérez and W. W. McGovern, The m-topology on $C_m(X)$ revisited, Topology Appl. 153 (2006), 1838-1848. https://doi.org/10.1016/j.topol.2005.06.016

A. Hayati, M. Namdari and M. Paimann, On countably uniform closed spaces, Quaestiones Mathematicae 42, no. 5 (2019), 593-604. https://doi.org/10.2989/16073606.2018.1476415

E. Hewitt, Rings of real-valued continuous functions, I, Trans. Amer. Math. Soc. 64 (1948), 45-99. https://doi.org/10.1090/S0002-9947-1948-0026239-9

O. A. S. Karamzadeh and Z. Keshtkar, On c-realcompact spaces, Quaestiones Mathematicae 41, no. 8 (2018), 1135-1167. https://doi.org/10.2989/16073606.2018.1441919

O. A. S. Karamzadeh, M. Namdari and S. Soltanpour, On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol. 16, no. 2 (2015), 183-207. https://doi.org/10.4995/agt.2015.3445

G. D. Maio, L. Hola, D. Holy and R. A. McCoy, Topologies on the space of continuous functions, Topology Appl. 86 (1998), 105-122. https://doi.org/10.1016/S0166-8641(97)00114-4

M. Namdari and A. Veisi, Rings of quotients of the subalgebra of C(X) consisting of functions with countable image, Inter. Math. Forum 7 (2012), 561-571.

J. R. Porter and R. G. Woods, Extensions and Absolutes of Hausdorff, Springer-Verlag, 1988. https://doi.org/10.1007/978-1-4612-3712-9

A. Veisi, $e_c$-Filters and $e_c$-ideals in the functionally countable subalgebra of $C^*(X)$, Appl. Gen. Topol. 20, no. 2 (2019), 395-405. https://doi.org/10.4995/agt.2019.11524

A. Veisi, On the $m_c$-topology on the functionally countable subalgebra of C(X), Journal of Algebraic Systems 9, no. 2 (2022), 335-345.

A. Veisi and A. Delbaznasab, Metric spaces related to Abelian groups, Appl. Gen. Topol. 22, no. 1 (2021), 169-181. https://doi.org/10.4995/agt.2021.14446