Recent zbMATH articles in MSC 54D20https://zbmath.org/atom/cc/54D202021-06-15T18:09:00+00:00WerkzeugOn \(\mathcal{I}\)-quotient mappings and \(\mathcal{I}\)-\(cs'\)-networks under a maximal ideal.https://zbmath.org/1460.540152021-06-15T18:09:00+00:00"Zhou, Xiangeng"https://zbmath.org/authors/?q=ai:zhou.xiangengIn the present paper, for an ideal \(I\) on \(\mathbb{N}\) and a mapping \(f:X\rightarrow Y\), the notions of \(I\)-quotient mapping and \(I\)-\(cs' \)-network for a topological space are introduced. Also, properties of the notions of \(I\)-quotient mappings and \(I\)-\(cs' \)-networks are studied.
Reviewer: Erdal Ekici (Çanakkale)On HČ-completeness and Katětov extensions.https://zbmath.org/1460.540162021-06-15T18:09:00+00:00"Martínez-Morales, J. A."https://zbmath.org/authors/?q=ai:martinez-morales.j-a"Rojas-Sánchez, A. D."https://zbmath.org/authors/?q=ai:rojas-sanchez.a-d"Tamariz-Mascarúa, Á."https://zbmath.org/authors/?q=ai:tamariz-mascarua.angelSome years after \textit{E. Čech} published [``On bicompact spaces'', Ann. Math. (2) 38, 823--844 (1937; Zbl 0017.42803)], several authors, e.g., see \textit{A. V. Arkhangel'skij} in [Vestn. Mosk. Univ., Ser. I 16, No. 2, 37--39 (1961; Zbl 0106.15702)] and \textit{Z. Frolík} in [Czech. Math. J. 10(85), 359--379 (1960; Zbl 0100.18701)], and later, \textit{K. Császár} in [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 25, 229--238 (1982; Zbl 0509.54018)] obtained internal characterizations of Čech-complete spaces, such as the following one: A Tychonoff space \(X\) is Čech-complete iff (*) there exists in \(X\) a sequence \(\{{\mathcal A}_n\}\) of open covers such that if \(\mathcal F\) is any centered family of closed subsets of \(X\), and there is for every \(n\in{\mathbb N}\) a set \(F_n\in{\mathcal F}\) contained in a set \(A_n\in {\mathcal A}_n\), then \(\bigcap\mathcal F\neq\emptyset\) (a family is said to be \textit{centered} iff it has the finite intersection property).
In the article under review its authors define a Hausdorff space \(X\) to be \textit{HČ-complete} (\textit{strongly HČ-complete}) iff it satisfies the condition (*) (iff there exists in \(X\) an open cover \(\mathcal A\) such that if \(\mathcal F\) is any centered family of closed subsets of \(X\), and there is for every \(n\in{\mathbb N}\) a set \(F_n\in\mathcal F\) contained in a set \(A_n\in\mathcal A\), then \(\bigcap{\mathcal F}\neq\emptyset\)). They state that their purpose is to continue the study of such spaces done in [Császár, loc. cit.] and in [\textit{D. K. McNeill}, Properties of \(H\)-sets, Katětov spaces and \(H\)-closed extensions with countable remainder. University of Kansas (Ph.D. Thesis) (2011)], and to focus especially on properties of a Hausdorff space \(X\) which ensure that its Katětov extension \(\kappa X\) be HČ-complete. We recall that \(\kappa X=X\cup\{\mathcal U:\mathcal U\text{ is a nonconvergent open ultrafilter on }X\}\), topologized so that a set \(V\subset\kappa X\) is defined to be \textit{open in \(\kappa X\)} iff (i) \(V\cap X\) is open in \(X\), and (ii) for every \({\mathcal U}\in \kappa X\setminus X\), if \(\mathcal U\in V\), then \(V\cap X\in\mathcal U\). Some of the examples and theorems they obtain are the following.
Locally compact Hausdorff implies strongly HČ-complete, which implies HČ-complete, and \({\mathbb R}\setminus{\mathbb Q}\) illustrates the two properties are distinct. Both properties are closed hereditary, there exists a Urysohn HČ-complete space which is not of the second category, but any HČ-complete space in which every nonempty open set contains the closure of a nonempty open set is Baire. If a Hausdorff space \(X= D\cup K\), where \(K\) is compact, \(D\) is discrete, and \(D\cap K=\emptyset\), then \(\kappa X\) is strongly HČ-complete. If an HČ-complete (strongly HČ-complete) space \(X\) either has a dense set of isolated points or is countably compact, then \(\kappa X\) is HČ-complete (strongly HČ-complete). If \(X\) is a space without isolated points and \(\kappa X\) is HČ-complete, then \(X\) must be feebly compact. Noting that Čech-completeness is known to be countably productive, the authors raise several questions about productivity and HČ-completeness, and they prove that for any positive integer \(m\) and infinite discrete space \(X\), the space \((\kappa X)^m\) is strongly HČ-complete. They also provide a new and different proof of the theorem of \textit{C.-T. Liu} [Trans. Am. Math. Soc. 130, 86--104 (1968; Zbl 0153.52301)] which gives necessary and sufficient conditions for a family of nonempty Hausdorff spaces \(\{X_\alpha\}\) that \(\kappa(\Pi X_\alpha)\) and \(\Pi\kappa X_\alpha\) be equivalent extension spaces.
Reviewer: Robert M. Stephenson Jr. (Columbia)A more balanced approach to ideal variation of \(\gamma\)-covers.https://zbmath.org/1460.540132021-06-15T18:09:00+00:00"Das, Pratulananda"https://zbmath.org/authors/?q=ai:das.pratulananda"Samanta, Upasana"https://zbmath.org/authors/?q=ai:samanta.upasana"Chandra, Debraj"https://zbmath.org/authors/?q=ai:chandra.debrajIn this note, the authors introduce the notion of \(G\)-\(\mathcal{I}\)-\(\gamma\) cover as a generalized and more balanced version of \(\mathcal{I}\)-\(\gamma\)-cover and study some of its basic selection properties as also certain Ramsey like properties and splittability properties. Certain characterizations of \(\gamma\)-sets are obtained in terms of \(G\)-\(\mathcal{I}\)-\(\gamma\) covers. The main results are the following:
\textbf{Theorem 1.} If \(X\) satisfies \(S_{fin}(\Omega, \mathcal{O}^{I-gp})\) then \(X\) satisfies \(U_{fin}(\mathcal{O}, G\textmd{-}\mathcal{I}\textmd{-}\Gamma)\) provided \(S_{1}(\mathcal{F}(I), \mathcal{F}(I))\) holds.
\textbf{Theorem 2.} For a space \(X\), \((\begin{array}{l} \ \ \ \Omega\\
G\textmd{-}\mathcal{I}\textmd{-}\Gamma \end{array})=S_{1}(\Omega, G\textmd{-}\mathcal{I}\textmd{-}\Gamma)\).
\textbf{Theorem 3.} For a space \(X\), \(S_{1}(\Omega, \Gamma)=S_{1}(G\textmd{-}\mathcal{I}\textmd{-}\Gamma, \Gamma)\).
Reviewer: Zuquan Li (Hangzhou)A note on \(\eta_1\)-spaces.https://zbmath.org/1460.540212021-06-15T18:09:00+00:00"Bennett, Harold"https://zbmath.org/authors/?q=ai:bennett.harold-r"Davis, Sheldon"https://zbmath.org/authors/?q=ai:davis.sheldon-w"Lutzer, David"https://zbmath.org/authors/?q=ai:lutzer.david-jFelix Hausdorff defined an \(\eta_1\)-set to be a linearly ordered set \((X, <)\) such that for any two countable subsets \(A, B\) with \(A < B\), the interval \((a,b)\) is non-empty for every \(a \in A, b \in B\). The authors study \(\eta_1\)-sets \(X\) with the open interval topology and with \(|X| = 2^\omega\), which they call small \(\eta_1\)-sets. In an extensive study of such sets, they include many nice results and examples addressing products, monotonically normality, and paracompactness, among other topics. The authors show that the Continuum Hypothesis is equivalent to each of the following: a) every small \(\eta_1\)-set is paracompact, b) every small \(\eta_1\)-set is realcompact, c) every small \(\eta_1\)-set \(X\) is homeomorphic to \(X^2\), and d) for every small \(\eta_1\)-set \(X\), \(X^2\) is normal. Several open questions are also presented.
Reviewer: Thomas Richmond (Bowling Green)Rothberger and Rothberger-type star selection principles on hyperspaces.https://zbmath.org/1460.540082021-06-15T18:09:00+00:00"Díaz-Reyes, Jesús"https://zbmath.org/authors/?q=ai:reyes.jesus-diaz"Ramírez-Páramo, Alejandro"https://zbmath.org/authors/?q=ai:ramirez-paramo.alejandro"Tenorio, Jesús F."https://zbmath.org/authors/?q=ai:tenorio.jesus-fFor a topological space \(X\), let \(CL (X)\) denote the hyperspace consisting of all nonempty closed subsets of \(X\) endowed with the Vietoris topology, and let \(\mathbb{K}(X)\) (resp., \(\mathbb{F}(X)\), \(\mathbb{CS}(X)\)) be the subspace consisting of all nonempty compact subsets (resp., nonempty finite subsets, convergent sequences) of \(X\). Motivated by the work of \textit{Z. Li} [Topology Appl. 212, 90--104 (2016; Zbl 1355.54014)], the authors characterize the Rothberger property and two selection principles called star-Rothberger and strongly star-Rothberger in the spaces \(CL (X)\), \(\mathbb{K}(X)\), \(\mathbb{F}(X)\) and \(\mathbb{CS}(X)\).
Let \(X\) be a topological space and let \(\Lambda\) be one of the hyperspaces \(CL (X)\), \(\mathbb{K}(X)\), \(\mathbb{F}(X)\) and \(\mathbb{CS}(X)\). Let \(\mathbf{S}_1(\mathcal{A},\mathcal{B})\) be the selection principle defined by \textit{M. Scheepers} [Topology Appl. 69, No. 1, 31--62 (1996; Zbl 0848.54018)]. The authors introduce the notion of \(\pi_V(\Lambda)\)-network modifying the notion of \(\pi_V\)-network defined by Li [loc. cit.], and two selection principles \(\mathbf{S}_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\) and \(\mathbf{S}^*_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\), where \(\Pi_V (\Lambda)\) is the collection of \(\pi_V(\Lambda)\)-networks of \(X\). The following theorems are proved: \(\Lambda\) has the Rothberger property if and only if \(X\) satisfies \(\mathbf{S}_{1} (\Pi_V(\Lambda),\Pi_V(\Lambda))\); \(\Lambda\) is strongly star-Rothberger if and only if \(X\) satisfies \(\mathbf{S}_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\); and \(\Lambda\) is star-Rothberger if and only if \(X\) satisfies \(\mathbf{S}^*_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\). Let \(\mathcal{D}\) denote the family of dense subsets of a given space. The authors also give a characterization of the selection principle \(\mathbf{S}_1 (\mathcal{D},\mathcal{D})\) for \(\Lambda\) following ideas by Li [loc. cit.].
Reviewer: Takamitsu Yamauchi (Matsuyama)Unbounded towers and products.https://zbmath.org/1460.540142021-06-15T18:09:00+00:00"Szewczak, Piotr"https://zbmath.org/authors/?q=ai:szewczak.p"Włudecka, Magdalena"https://zbmath.org/authors/?q=ai:wludecka.magdalenaA \(\kappa\)-unbounded tower is a sequence \(\langle x_\alpha:\alpha<\kappa\rangle\) of infinite subsets of~\(\mathbb{N}\) that is decreasing mod~finite and has the property that its set of counting functions is unbounded mod~finite in~\(\mathbb{N}^\mathbb{N}\). A \(\kappa\)-generalized tower is a family, \(X\) of infinite subsets of~\(\mathbb{N}\) of cardinality at least~\(\kappa\) such that for every infinite subset~\(a\) of~\(\mathbb{N}\) there is an infinite subset~\(b\) of~\(\mathbb{N}\) such that the set of~\(x\in X\) for which \(x\cap\bigcup_{n\in b}[a_n,a_{n+1})\) is infinite has cardinality less than~\(\kappa\). These sets are considered with the subspace topology they inherit from the Cantor set.
The authors consider selection properties, notably \(S_1(\Gamma,\Gamma)\) and \(S_1(\Gamma_\mathrm{Bor},\Gamma_\mathrm{Bor})\) [\textit{M. Scheepers}, Topology Appl. 69, No. 1, 31--62 (1996; Zbl 0848.54018)]. They show that the product of finitely many \(\mathfrak{b}\)-generalized towers and one subset of~\(\mathbb{R}\) that satisfies \(S_1(\Gamma_\mathrm{Bor},\Gamma_\mathrm{Bor})\) satisfies \(S_1(\Gamma,\Gamma)\). Similar results are obtained for \(\mathfrak{p}\)-generalized towers and the property~\(\binom\Omega\Gamma\) (every \(\omega\)-cover contains a \(\gamma\)-cover).
Reviewer: K. P. Hart (Delft)Star versions of the Rothberger property on hyperspaces.https://zbmath.org/1460.540072021-06-15T18:09:00+00:00"Casas-de la Rosa, Javier"https://zbmath.org/authors/?q=ai:casas-de-la-rosa.javier"Martínez-Ruiz, Iván"https://zbmath.org/authors/?q=ai:martinez-ruiz.ivan"Ramírez-Páramo, Alejandro"https://zbmath.org/authors/?q=ai:ramirez-paramo.alejandroA space \(X\) is said to be star-Rothberger (resp., strongly star-Rothberger) if for each sequence \((\mathcal U_n)\) of open covers of \(X\) there is a sequence \((U_n)\) (resp., \((x_n)\)) such that for each \(n\), \(U_n \in \mathcal U_n\) (resp., \(x_n \in X\)) and \(\{St(U_n,\mathcal U_n): n \in \mathbb N\}\) (resp., \(\{St(x_n,\mathcal U_n): n \in \mathbb N\}\)) is an open cover of \(X\). The authors characterize the star-Rothberger and strongly star-Rothberger properties of the hyperspace over a space \(X\) equipped with the Fell topology in terms of selective properties of \(X\) of \(\pi\)-network-type. They also characterize the strongly star-Rothberger property on hyperspaces endowed with the lower Vietoris topology. Selective versions of metacompactness and and mesocompactness are also considered.
Reviewer: Ljubiša D. Kočinac (Niš)