Super properties and net weight.

*(English)*Zbl 1444.54017In what follows, all topological spaces are \(T_3\) (i.e., Hausdorff and regular) and we assume the reader is familiar with Martin’s Axiom, \(\mathbf{MA}(\kappa)\), and with the Continuum Hypothesis, \(\mathbf{CH}\). The cardinality of a set \(X\) is denoted by \(|X|\).

A network for a topological space \(X\) is a family \(\mathcal{N}\) of (non-necessarily open) subsets of \(X\) such that every open set of \(X\) is the union of a subfamily of \(\mathcal{N}\). The net weight of \(X\), denoted by \(nw(X)\), is the least cardinality of a network for \(X\). Every base is a network and \(\{\{x\}: x \in X\}\) is a network, so clearly \(nw(X) \leqslant \min\{|X|,w(X)\}\) – where \(w(X)\) (the weight of \(X\)) is the least cardinality of a base of \(X\).

Given a topological space \(X\) and a cardinal \(\kappa\), a \(\kappa\)-assignment for \(X\) is a sequence \(\mathcal{U} = \{(x_\alpha,U_\alpha): \alpha < \kappa\}\), where each \(U_\alpha\) is open in \(X\) and \(x_\alpha \in U_\alpha\). An assignment for \(X\) is an \(\omega_1\)-assignment. A topological space \(X\) is said to be HG (Hereditarily Good) if for all assignments \(\mathcal{U}\) of \(X\) there are \(\alpha \neq \beta\) such that \([x_\beta \in U_\alpha\,\, \&\,\, x_\alpha \in U_\beta]\). The property HG was introduced in [J. E. Hart and K. Kunen, Topol. Proc. 55, 147–174 (2020; Zbl 1444.54016)] and it is a natural strengthening of the properties \(HS\) (i.e. Hereditarily Separable) and \(HL\) (i.e. Hereditarily Lindelöf).

A topological space \(X\) is strongly HG, or \(\text{st}\)HG, if all finite powers of \(X\) are HG (equivalently, \(X^\omega\) is HG), and it is said to be super HG, or \(\text{su}\)HG, if for all assignments for \(X\) there is some \(S\subseteq\omega_1\) with \(|S| = \aleph_1\) such that for all \(\alpha, \beta \in S\) one has \([x_\beta \in U_\alpha\,\, \&\,\, x_\alpha \in U_\beta]\). \(\text{su}\)HG trivially implies HG and, as finite products of \(\text{su}\)HG spaces are \(\text{su}\)HG spaces, one also has that \(\text{su}\)HG implies \(\text{st}\)HG. It was previously shown by the authors in [loc. cit.] that \(\mathbf{CH}\) produces an example of a \(\text{st}\)HG space that is not \(\text{su}\)HG; however, in the very same paper it was established that MA\((\aleph_1)\) implies that every \(\text{st}\)HG space is \(\text{su}\)HG.

In the paper under review, the authors proceed with the investigation of super properties. The main results of the paper are the following:

(i) \(\mathbf{MA}(\kappa)\) implies that every \(\text{st}\)HG space with \(|X| \leqslant \kappa\) and \(w(X) \leqslant \kappa\) has countable net weight.

The paper includes a brief proof of the above result and remarks that it is essentially a consequence of Theorem 2.1 of [I. Juhász et al., Commentat. Math. Univ. Carol. 37, No. 1, 159–170 (1996; Zbl 0862.54003)].

(ii) If \(nw(X) \leqslant \aleph_0\) then \(X\) is \(\text{su}\)HG.

The above result is an easy consequence of a pigeonhole argument. It follows that spaces with countable net weight are trivial examples of \(\text{su}\)HG spaces. Nevertheless, in the paper also the following is proved:

(iii) It is consistent with \(\mathbf{ZFC}\) to have MA and \(\mathfrak{c}=\aleph_2\) and a first countable \(\text{su}\)HG space \(X\) with \(|X| = w(X) = nw(X) = \aleph_2\).

A network for a topological space \(X\) is a family \(\mathcal{N}\) of (non-necessarily open) subsets of \(X\) such that every open set of \(X\) is the union of a subfamily of \(\mathcal{N}\). The net weight of \(X\), denoted by \(nw(X)\), is the least cardinality of a network for \(X\). Every base is a network and \(\{\{x\}: x \in X\}\) is a network, so clearly \(nw(X) \leqslant \min\{|X|,w(X)\}\) – where \(w(X)\) (the weight of \(X\)) is the least cardinality of a base of \(X\).

Given a topological space \(X\) and a cardinal \(\kappa\), a \(\kappa\)-assignment for \(X\) is a sequence \(\mathcal{U} = \{(x_\alpha,U_\alpha): \alpha < \kappa\}\), where each \(U_\alpha\) is open in \(X\) and \(x_\alpha \in U_\alpha\). An assignment for \(X\) is an \(\omega_1\)-assignment. A topological space \(X\) is said to be HG (Hereditarily Good) if for all assignments \(\mathcal{U}\) of \(X\) there are \(\alpha \neq \beta\) such that \([x_\beta \in U_\alpha\,\, \&\,\, x_\alpha \in U_\beta]\). The property HG was introduced in [J. E. Hart and K. Kunen, Topol. Proc. 55, 147–174 (2020; Zbl 1444.54016)] and it is a natural strengthening of the properties \(HS\) (i.e. Hereditarily Separable) and \(HL\) (i.e. Hereditarily Lindelöf).

A topological space \(X\) is strongly HG, or \(\text{st}\)HG, if all finite powers of \(X\) are HG (equivalently, \(X^\omega\) is HG), and it is said to be super HG, or \(\text{su}\)HG, if for all assignments for \(X\) there is some \(S\subseteq\omega_1\) with \(|S| = \aleph_1\) such that for all \(\alpha, \beta \in S\) one has \([x_\beta \in U_\alpha\,\, \&\,\, x_\alpha \in U_\beta]\). \(\text{su}\)HG trivially implies HG and, as finite products of \(\text{su}\)HG spaces are \(\text{su}\)HG spaces, one also has that \(\text{su}\)HG implies \(\text{st}\)HG. It was previously shown by the authors in [loc. cit.] that \(\mathbf{CH}\) produces an example of a \(\text{st}\)HG space that is not \(\text{su}\)HG; however, in the very same paper it was established that MA\((\aleph_1)\) implies that every \(\text{st}\)HG space is \(\text{su}\)HG.

In the paper under review, the authors proceed with the investigation of super properties. The main results of the paper are the following:

(i) \(\mathbf{MA}(\kappa)\) implies that every \(\text{st}\)HG space with \(|X| \leqslant \kappa\) and \(w(X) \leqslant \kappa\) has countable net weight.

The paper includes a brief proof of the above result and remarks that it is essentially a consequence of Theorem 2.1 of [I. Juhász et al., Commentat. Math. Univ. Carol. 37, No. 1, 159–170 (1996; Zbl 0862.54003)].

(ii) If \(nw(X) \leqslant \aleph_0\) then \(X\) is \(\text{su}\)HG.

The above result is an easy consequence of a pigeonhole argument. It follows that spaces with countable net weight are trivial examples of \(\text{su}\)HG spaces. Nevertheless, in the paper also the following is proved:

(iii) It is consistent with \(\mathbf{ZFC}\) to have MA and \(\mathfrak{c}=\aleph_2\) and a first countable \(\text{su}\)HG space \(X\) with \(|X| = w(X) = nw(X) = \aleph_2\).

Reviewer: Samuel Gomes da Silva (Salvador)

##### MSC:

54D70 | Base properties of topological spaces |

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |

54A35 | Consistency and independence results in general topology |

PDF
BibTeX
XML
Cite

\textit{J. E. Hart} and \textit{K. Kunen}, Topology Appl. 274, Article ID 107144, 19 p. (2020; Zbl 1444.54017)

Full Text:
DOI

##### References:

[1] | Gruenhage, G., Cosmicity of cometrizable spaces, Trans. Am. Math. Soc., 313, 301-315 (1989) · Zbl 0667.54012 |

[2] | Hajnal, A.; Juhász, I., Weakly separated subspaces and networks, (Logic Colloquium ’78. Logic Colloquium ’78, Mons, 1978 (1979), North-Holland: North-Holland Amsterdam), 235-245 |

[3] | Hart, J. E.; Kunen, K., Spaces with no S or L subspaces, Topol. Proc., 55, 147-174 (2020) |

[4] | Hart, J. E.; Kunen, K., Properties and super properties, Topol. Proc., 55, 175-185 (2020) |

[5] | Heath, R. W., On certain first countable spaces, (Topology Seminar Wisconsin 1965. Topology Seminar Wisconsin 1965, Ann. of Math. Studies, vol. 60 (1966)), 103-113 · Zbl 0147.41603 |

[6] | Jech, T., Set Theory (2002), Springer |

[7] | Juhász, I., Cardinal Functions in Topology (1975), Mathematisch Centrum |

[8] | Juhász, I., HFD and HFC type spaces, with applications, Topol. Appl., 126, 217-262 (2002) · Zbl 1012.54003 |

[9] | Juhász, I.; Soukup, L.; Szentmiklóssy, Z., Forcing countable networks for spaces satisfying \(R( X^\omega) = \omega \), Comment. Math. Univ. Carol., 37, 159-170 (1996) · Zbl 0862.54003 |

[10] | Kunen, K., Random and Cohen reals, (Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 887-911 |

[11] | Kunen, K., Set Theory (2011), College Publications · Zbl 1262.03001 |

[12] | Michael, E. A., Paracompactness and the Lindelöf property in finite and countable Cartesian products, Compos. Math., 23, 199-214 (1971) · Zbl 0216.44304 |

[13] | Roitman, J., Basic S and L, (Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 295-326 |

[14] | Tkačenko, M. G., Chains and cardinals, Dokl. Akad. Nauk SSSR, 239, 546-549 (1978), (in Russian) |

[15] | Todorčević, S., Partition Problems in Topology, Contemporary Mathematics, vol. 84 (1989), American Mathematical Society · Zbl 0659.54001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.