Some classes of topological spaces related to zero-sets
Keywords:zero-set, almost P-space, compact space, z-embedded subset
An almost P-space is a topological space in which every zero-set is regular-closed. We introduce a large class of spaces, C-almost P-space (briefly CAP-space), consisting of those spaces in which the closure of the interior of every zero-set is a zero-set. In this paper we study CAP-spaces. It is proved that if X is a dense and Z#-embedded subspace of a space T, then T is CAP if and only if X is a CAP and CRZ-extended in T (i.e, for each regular-closed zero-set Z in X, clTZ is a zero-set in T). In 6P.5 of  it was shown that a closed countable union of zero-sets need not be a zero-set. We call X a CZ-space whenever the closure of any countable union of zero-sets is a zero-set. This class of spaces contains the class of P-spaces, perfectly normal spaces, and is contained in the cozero complemented spaces and CAP-spaces. In this paper we study topological properties of CZ (resp. cozero complemented)-space and other classes of topological spaces near to them. Some algebraic and topological equivalent conditions of CZ (resp. cozero complemented)-space are characterized. Examples are provided to illustrate and delimit our results.
F. Azarpanah, On almost P-spaces, Far East J. Math. Sci. 1 (2000), 121-132.
F. Azarpanah, M. Ghirati and A. Taherifar, Closed ideals in C(X) with different reperesentations, Houston Journal of Mathematics 44, no. 1 (2018), 363-383.
F. Azarpanah, A. A. Hesari, A. R. Salehi and A. Taherifar, A Lindelöfication, Topology Appl. 245 (2018), l46-61. https://doi.org/10.1016/j.topol.2018.06.009
F. Azarpanah and M. Karavan, On nonregular ideals and $z^o$-ideals in C(X), Czechoslovak Math. J. 55 (2005), 397-407. https://doi.org/10.1007/s10587-005-0030-0
F. Azarpanah, O. A. S. Karamzadeh and A. Rezai Aliabad, On z-ideals in C(X), Fund. Math. 160 (1999), 15-25. https://doi.org/10.4064/fm_1999_160_1_1_15_25
R. L. Blair and A. W. Hager, Extension of zero-sets and real-valued functions, Math. Z. 136 (1974), 41-52. https://doi.org/10.1007/BF01189255
F. Dashiel, A. Hager and M. Henriksen, Order-Cauchy completions and vector lattices of continuous functions, Canad. J. Math. XXXII (1980), 657-685. https://doi.org/10.4153/CJM-1980-052-0
L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976.
M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130. https://doi.org/10.1090/S0002-9947-1965-0194880-9
M. Henriksen and G. Woods, Cozero complemented spaces; when the space of minimal prime ideals of a C(X) is compact, Topology Appl. 141 (2004), 147-170. https://doi.org/10.1016/j.topol.2003.12.004
R. Levy, Almost P-spaces, Canad. J. Math. 2 (1977), 284-288. https://doi.org/10.4153/CJM-1977-030-7
A. Taherifar, Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84-97. https://doi.org/10.1016/j.topol.2014.01.017
A. I. Veksler, P'-points, P'-sets, P'-spaces. A new class of order-continuous measures and functionals, Sov. Math. Dokl. 14 (1973), 1445-1450.
How to Cite
Copyright (c) 2022 Applied General Topology
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.