Recent zbMATH articles in MSC 54Chttps://zbmath.org/atom/cc/54C2021-06-15T18:09:00+00:00WerkzeugCoincidence results for compositions of multivalued maps based on countable compactness principles.https://zbmath.org/1460.540512021-06-15T18:09:00+00:00"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: We present a general Mönch coincidence type result for set-valued maps on Hausdorff topological spaces.
Reviewer: Reviewer (Berlin)On \(\mathcal{I}\)-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 the sum of \(z\)-ideals in subrings of \(C(X)\).https://zbmath.org/1460.540122021-06-15T18:09:00+00:00"Azarpanah, Fariborz"https://zbmath.org/authors/?q=ai:azarpanah.f"Parsinia, Mehdi"https://zbmath.org/authors/?q=ai:parsinia.mehdiThroughout, \(X\) is a completely regular Hausdorff space, \(C(X)\) is the ring of real-valued continuous functions on \(X\), and \(C^*(X)\) is the subring of \(C(X)\) consisting of bounded functions. A subalgebra \(A(X)\) of \(C(X)\) is called intermediate if \(C^*(X)\subseteq A(X)\subseteq C(X)\). A simple but useful proposition that sets the stage in the paper is that the intermediate subalgebras of \(C(X)\) are precisely its absolutely convex subalgebras. An ideal in a commutative ring is called a \(z\)-ideal if the intersection of all maximal ideals containing any member of the ideal is contained in the ideal. In \(C(X)\) this is equivalent to saying that any function with the same zero-set as a member of the ideal is a member of the ideal. Modifying this, the authors of the paper under review say an ideal \(I\) of a subalgebra \(A(X)\) is called a \(z_A\)-ideal in case whenever \(Z(f)\subseteq Z(g)\) with \(f\in I\) and \(g\in A\), then \(g\in I\). They show that the \(z\)-ideals of \(A(X)\) are precisely its \(z_A\)-ideals if and only if \(A(X)=C(X)\). They also show that in any intermediate subalgebra, the sum of two \(z_A\)-ideals is a \(z_A\)-ideal or the entire ring; a result reminiscent of the fact that in \(C(X)\) the sum of two \(z\)-ideals is a \(z\)-ideal or the entire ring.
The major achievement (among several other significant results) in the second part of the paper is the characterization of the spaces \(X\) such that, for any ideal \(I\) of \(C(X)\), the sum of two \(z\)-ideals in the subring \(I+\mathbb R\) of \(C(X)\) is a \(z\)-ideal or the whole of \(I+\mathbb R\). They are precisely the \(F\)-spaces.
Reviewer: Themba Dube (Unisa)Further discussion about transitivity and mixing of continuous maps on compact metric spaces.https://zbmath.org/1460.370072021-06-15T18:09:00+00:00"Liu, Guo"https://zbmath.org/authors/?q=ai:liu.guo"Lu, Tianxiu"https://zbmath.org/authors/?q=ai:lu.tianxiu"Yang, Xiaofang"https://zbmath.org/authors/?q=ai:yang.xiaofang"Waseem, Anwar"https://zbmath.org/authors/?q=ai:waseem.anwarFor a compact metric space \((X,d)\) and a continuous map \(f:X\rightarrow X\), the aim of the paper is to study the relationships among various forms of transitivity, such as syndetical transitivity, total transitivity and strong transitivity. Necessary and sufficient conditions for \(f\) to be totally, strongly, syndetically transitive, weakly mixing, mixing, and minimal are investigated.
Reviewer: Erdal Ekici (Çanakkale)Fuzzy fixed point theorems in ordered cone metric spaces.https://zbmath.org/1460.540352021-06-15T18:09:00+00:00"Azam, Akbar"https://zbmath.org/authors/?q=ai:azam.akbar"Mehmood, Nayyar"https://zbmath.org/authors/?q=ai:mehmood.nayyar"Rashid, Maliha"https://zbmath.org/authors/?q=ai:rashid.maliha"Radenović, Stojan"https://zbmath.org/authors/?q=ai:radenovic.stojanSummary: In this article, we introduce the notion of the multivalued fuzzy mappings satisfying w.l.b property and l.b properties and prove some results for multivalued generalized contractive fuzzy mappings in ordered-cone metric spaces without the assumption of normality on cones. We generalize many results in the literature.
Reviewer: Reviewer (Berlin)\(\psi^*\)-closed sets in fuzzy topological spaces.https://zbmath.org/1460.540042021-06-15T18:09:00+00:00"Allah, M. A. Abd"https://zbmath.org/authors/?q=ai:allah.m-a-abd"Nawar, A. S."https://zbmath.org/authors/?q=ai:nawar.ashraf-sSummary: In this paper, we introduce a new class of fuzzy sets, namely, fuzzy \(\psi^*\)-closed sets for fuzzy topological spaces, and some of their properties have been proved. Further, we introduce fuzzy \(\psi^*\)-continuous, fuzzy \(\psi^*\)-irresolute functions, and fuzzy \(\psi^*\)-closed (open) functions, as applications of these fuzzy sets, fuzzy \(T_{1/5}\)-spaces, fuzzy \({T}_{1/5}^{\psi \ast}\)-spaces, and fuzzy \(^{\psi *} T_{1/5}\)-spaces.
Reviewer: Reviewer (Berlin)Coincidence points for mappings under generalized contraction.https://zbmath.org/1460.540412021-06-15T18:09:00+00:00"Gairola, Ajay"https://zbmath.org/authors/?q=ai:gairola.ajay"Joshi, Mahesh C."https://zbmath.org/authors/?q=ai:joshi.mahesh-chandraSummary: In this paper we establish some results on the existence of coincidence and fixed points for multi-valued and single-valued mappings extending the result of \textit{Y.-Q. Feng} and \textit{S.-Y. Liu} [J. Math. Anal. Appl. 317, No. 1, 103--112 (2006; Zbl 1094.47049)] and \textit{Z.-Q. Liu} et al. [Fixed Point Theory Appl. 2010, Article ID 870980, 18 p. (2010; Zbl 1207.54062)]. It is also proved by counterexample that our results generalize and extend some well-known results.
Reviewer: Reviewer (Berlin)Stationary points of set-valued contractive and nonexpansive mappings on ultrametric spaces.https://zbmath.org/1460.540442021-06-15T18:09:00+00:00"Hosseini, Meraj"https://zbmath.org/authors/?q=ai:hosseini.meraj"Nourouzi, Kourosh"https://zbmath.org/authors/?q=ai:nourouzi.kourosh"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, we show that contractive set-valued mappings on spherically complete ultrametric spaces have stationary (or end) points if they have the approximate stationary point property. We also extend some known fixed point results to nonexpansive set-valued mappings.
Reviewer: Reviewer (Berlin)Distinguished \(C_p (X)\) spaces.https://zbmath.org/1460.540112021-06-15T18:09:00+00:00"Ferrando, J. C."https://zbmath.org/authors/?q=ai:ferrando.juan-carlos"Kąkol, J."https://zbmath.org/authors/?q=ai:kakol.jerzy"Leiderman, A."https://zbmath.org/authors/?q=ai:leiderman.arkady-g"Saxon, S. A."https://zbmath.org/authors/?q=ai:saxon.stephen-aThe authors continue the study of distinguished \(C_p(X)\) spaces, where \(X\) is an infinite Tychonoff space and \(C_p(X)\) is the vector space of real continuous functions defined on \(X\) equipped with the pointwise topology. They prove several results implying that these spaces are distinguished or not. Some typical results are the following: (1) \(C_p(X)\) is distinguished if and only if the strong dual \(L_{\beta}(X)\) is flat and if and only if the strong bidual is the product space \(\mathbb R^X\); (2) Let \(X\) be a Corson compact space, i.e., homeomorphic to a compact subset of a Banach space; then \(C_p(X)\) is distinguished if and only if \(X\) is scattered (each of its closed non-empty subspaces has an isolated point in the relative topology) and Eberlein compact (homeomorphic to a weakly compact subset of a Banach space).
Reviewer: Zoran Kadelburg (Beograd)Some nano topological structures via ideals and graphs.https://zbmath.org/1460.540092021-06-15T18:09:00+00:00"El-Atik, Abd El-Fattah A."https://zbmath.org/authors/?q=ai:el-atik.abd-el-fattah-a"Hassan, Hanan Z."https://zbmath.org/authors/?q=ai:hassan.hanan-zSummary: In this paper, new forms of nano continuous functions in terms of the notion of nano \(I \alpha\)-open sets called nano \(I \alpha\)-continuous functions, strongly nano \(I \alpha\)-continuous functions and nano \(I \alpha\)-irresolute functions will be introduced and studied. We establish new types of nano \(I \alpha\)-open functions, nano \(I \alpha\)-closed functions and nano \(I \alpha\)-homeomorphisms. A comparison between these types of functions and other forms of continuity will be discussed. We prove the isomorphism between simple graphs via the nano continuity between them. Finally, we apply these topological results on some models for medicine and physics which will be used to give a solution for some real-life problems.
Reviewer: Reviewer (Berlin)Coupled fixed points of monotone mappings in a metric space with a graph.https://zbmath.org/1460.540302021-06-15T18:09:00+00:00"Alfuraidan, Monther Rashed"https://zbmath.org/authors/?q=ai:alfuraidan.monther-rashed"Khamsi, Mohamed Amine"https://zbmath.org/authors/?q=ai:khamsi.mohamed-amineSummary: In this work, we define the concept of mixed $G$-monotone mappings defined on a metric space endowed with a graph. Then we obtain sufficient conditions for the existence of coupled fixed points for such mappings when a weak contractivity type condition is satisfied.
Reviewer: Reviewer (Berlin)Existence of best proximity points for set-valued cyclic Meir-Keeler contractions.https://zbmath.org/1460.540402021-06-15T18:09:00+00:00"Fakhar, Majid"https://zbmath.org/authors/?q=ai:fakhar.majid"Soltani, Zeinab"https://zbmath.org/authors/?q=ai:soltani.zeinab"Zafarani, Jafar"https://zbmath.org/authors/?q=ai:zafarani.jafarSummary: In this paper the concept of set-valued cyclic Meir-Keeler contraction map is introduced. The existence of best proximity point for such maps on a metric space with the UC property is presented.
Reviewer: Reviewer (Berlin)On the Cauchy problem for implicit differential equations with discontinuous right-hand side.https://zbmath.org/1460.340202021-06-15T18:09:00+00:00"Cubiotti, Paolo"https://zbmath.org/authors/?q=ai:cubiotti.paoloSummary: Given \(T>0\), a set \(Y\subseteq\mathbb{R}^n\), a point \(\xi\in\mathbb{R}^n\) and two functions \(f:[0,T]\times\mathbb{R}^n\to\mathbb{R}\) and \(g:Y\to\mathbb{R}\), we are interested in the Cauchy problem \(g(u')=f(t,u)\) in \([0,T]\), \(u(0)=\xi\). We prove an existence result for generalized solutions of the above problem, where the continuity of \(f\) with respect to the second variable is not assumed. As a matter of fact, a function \(f(t,x)\) satisfying our assumptions could be discontinuous (with respect to \(x\)) even at all points \(x\in\mathbb{R}^n\). As regards \(g\), we only require that it is continuous and locally nonconstant.
We also investigate the dependence of the solution set from the initial point \(\xi\). In particular, we give conditions under which the solution multifunction \(\mathcal{S}(\xi)\) admits an upper semicontinuous and compact-valued multivalued selection.
Reviewer: Reviewer (Berlin)On embedding of multidimensional Morse-Smale diffeomorphisms into topological flows.https://zbmath.org/1460.370192021-06-15T18:09:00+00:00"Grines, V."https://zbmath.org/authors/?q=ai:grines.vyacheslav-z"Gurevich, E."https://zbmath.org/authors/?q=ai:gurevich.e-a|gurevich.elena-ya"Pochinka, O."https://zbmath.org/authors/?q=ai:pochinka.olga-vSummary: \textit{J. Palis} [Topology 8, 385--405 (1969; Zbl 0189.23902)]
found necessary conditions for a Morse-Smale diffeomorphism on a closed \(n\)-dimensional manifold \(M^n\) to embed into a topological flow and proved that these conditions are also sufficient for \(n=2\). For the case \(n=3\) a possibility of wild embedding of closures of separatrices of saddles is an additional obstacle for Morse-Smale cascades to embed into topological flows. In this paper we show that there are no such obstructions for Morse-Smale diffeomorphisms without heteroclinic intersection given on the sphere \(S^n\), \(n\geq 4\), and Palis conditions again are sufficient for such diffeomorphisms.
Reviewer: Reviewer (Berlin)Generalized \(w\) closed sets in biweak structure spaces.https://zbmath.org/1460.540022021-06-15T18:09:00+00:00"Abu-Donia, Hassan Mustafa"https://zbmath.org/authors/?q=ai:abu-donia.hassan-mustafa"Hosny, Rodyna A."https://zbmath.org/authors/?q=ai:hosny.rodyna-aSummary: As a generalization of the classes of \(gw\) closed (resp. \(gw\) open, \(sgw\) closed) sets in a weak structure space \((X, w)\), the notions of \(ij\)-generalized \(w\) closed (resp. \(ij\)-generalized \(w\) open, \(ij\)-strongly generalized \(w\) closed) sets in a biweak structure space \((X, w_1, w_2)\) are introduced. In terms of these concepts, new forms of continuous functions between biweak spaces are constructed. Additionally, the concepts of \(ij\)-\(w\) normal, \(ij\)-\(gw\) normal, \(ij\)-\(wT_{\frac{1}{2}}\), and \(ij\)-\(w^{\sigma} T_{\frac{1}{2}}\) spaces are studied and several characterizations of them are acquired.
Reviewer: Reviewer (Berlin)On linear equations with polynomial coefficients.https://zbmath.org/1460.260182021-06-15T18:09:00+00:00"Kucharz, Wojciech"https://zbmath.org/authors/?q=ai:kucharz.wojciechSummary: We give a short survey of result on continuous (resp. continuous semialgebraic or regulous) solutions of linear equations with polynomial coefficients.
For the entire collection see [Zbl 1460.13001].
Reviewer: Reviewer (Berlin)Planar embeddings of Minc's continuum and generalizations.https://zbmath.org/1460.540102021-06-15T18:09:00+00:00"Anušić, Ana"https://zbmath.org/authors/?q=ai:anusic.ana\textit{S. B. Nadler jun.} and \textit{J. Quinn} [Embeddability and structure properties of real curves. Providence, RI: American Mathematical Society (AMS) (1972; Zbl 0236.54029)] asked whether for every point \(x\) of a chainable continuum \(X\) there is a planar embedding of \(X\) for which \(x\) is accessible. One of the candidates to find a counterexample was a continuum \(X_M\) constructed by Minc, see [Lect. Notes Pure Appl. Math. 230, 331--339 (2002; Zbl 1040.54500)], and the main result here gives a positive answer to the question of Nadler and Quinn for a class of continua including \(X_M\). The class of continua studied here are inverse limits \(I\overset{f}{\leftarrow}I\overset{f}{\leftarrow}\cdots\) of intervals with bonding map \(f\) a piecewise monotone post-critically finite locally eventually onto map. For this class it is shown that for every point there is a planar embedding that renders the point accessible.
Reviewer: Thomas B. Ward (Leeds)Various examples of the KKM spaces.https://zbmath.org/1460.490162021-06-15T18:09:00+00:00"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieSummary: Recall that a partial KKM space is a topological space satisfying an abstract form of the well-known KKM theorem, and a KKM space is a partial KKM space satisfying the `open-valued' version of the form. Recently, we have found several new examples of such spaces. As a continuation of the preceding two works, we are going to introduce old and new examples of the KKM spaces, which play a major role in applications of the KKM theory. Finally, we state why we should use triples \((X,D;\Gamma)\) for abstract convex spaces by giving further examples, and introduce an example of a partial KKM space which is not a KKM space.
Reviewer: Reviewer (Berlin)Paratopological groups: local versus global.https://zbmath.org/1460.540252021-06-15T18:09:00+00:00"Li, Piyu"https://zbmath.org/authors/?q=ai:li.piyu"Mou, Lei"https://zbmath.org/authors/?q=ai:mou.lei"Xu, Yanqin"https://zbmath.org/authors/?q=ai:xu.yanqinAn example is given in order to show that locally metrizable paratopological groups might be neither paracompact nor normal. This provides negative answers to questions previously posed in the field by \textit{A. Arhangel'skii} and \textit{M. Tkachenko} [Topological groups and related structures. Hackensack, NJ: World Scientific; Paris: Atlantis Press (2008; Zbl 1323.22001)] for paratopological groups. The authors also consider the conditions under which 2-pseudocompactness is a three space property in paratopological groups.
Reviewer: Watchareepan Atiponrat (Chiang Mai)Existence theorems of a new set-valued MT-contraction in \(b\)-metric spaces endowed with graphs and applications.https://zbmath.org/1460.540602021-06-15T18:09:00+00:00"Tiammee, Jukrapong"https://zbmath.org/authors/?q=ai:tiammee.jukrapong"Suantai, Suthep"https://zbmath.org/authors/?q=ai:suantai.suthep"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: A new concept of set-valued Mizoguchi-Takahashi G-contractions is introduced in this paper and some fixed point theorems for such mappings in b-metric spaces endowed with directed graphs are established under some sufficient conditions. Our results improve and extend those of [\textit{N. Mizoguchi} and \textit{W. Takahashi}, J. Math. Anal. Appl. 141, No. 1, 177--188 (1989; Zbl 0688.54028); \textit{A. Sultana} and \textit{V. Vetrivel}, J. Math. Anal. Appl. 417, No. 1, 336--344 (2014; Zbl 1368.54030)]. We also give some examples supporting our main results. As an applications, we prove the existence of fixed points for multivalued mappings satisfying generalized MT-contractive condition in \(\epsilon\)-chainable \(b\)-metric spaces and the existence of a solution for some integral equations.
Reviewer: Reviewer (Berlin)On certain versions of straightness.https://zbmath.org/1460.540182021-06-15T18:09:00+00:00"Das, Pratulananda"https://zbmath.org/authors/?q=ai:das.pratulananda"Pal, Sudip Kumar"https://zbmath.org/authors/?q=ai:pal.sudip-kumar"Adhikary, Nayan"https://zbmath.org/authors/?q=ai:adhikary.nayanAll spaces in the sequel are metric and \(C(X)\) denotes the set of all continuous real-valued functions on \(X\). \textit{A. Berarducci} et al. [Topology Appl. 146--147, 339--352 (2005; Zbl 1078.54014)] introduced and studied the notion of straight spaces: A space \(X\) is \textit{straight} if whenever \(X\) is the union of two closed sets, then \(f\in C(X)\) is uniformly continuous iff its restriction to each of the closed sets is uniformly continuous.
In this paper, the authors consider notions of continuity which are strictly weaker than the notion of uniform continuity, and introduce three versions of straightness, namely pre-straightness, pre\((\ast)\)-straightness and \(W\)-straightness: A space \(X\) is \textit{pre-straight} if whenever \(X\) is the union of closed sets, then \(f\in C(X)\) is Cauchy regular iff its restriction to each of the closed sets is Cauchy regular, where a mapping \(f:X\to Y\) is Cauchy regular if for any Cauchy sequence \(\{x_n\}\) in \(X\), \(\{f(x_n)\}\) is Cauchy in \(Y\). A space \(X\) is \textit{pre\((\ast)\)-straight} if whenever \(X\) is the union of closed sets, then a real-valued Cauchy regular function \(f\) is uniformly continuous iff its restriction to each of the closed sets is uniformly continuous. Recall that a sequence \(\{x_n\}\) in a metric space \((X, d)\) is \textit{quasi-Cauchy} if given \(\varepsilon>0\), there is a natural number \(n_0\) such that \(d(x_{n+1},x_n)<\varepsilon\) for all \(n\geq n_0\). A space \(X\) is \textit{\(W\)-straight} if whenever \(X\) is the union of closed sets, then \(f\in C(X)\) is ward continuous iff its restriction to each closed sets is ward continuous, where a mapping \(f: X\to Y\) is \textit{ward continuous} if it preserves quasi-Cauchy sequences.
The authors investigate some properties of these spaces and the relationship between these notions. For instance, they obtain the following: A space \(X\) is straight if and only if \(X\) is pre-straight and pre\((\ast)\)-straight. A space \(X\) is pre\((\ast)\)-straight if and only if its completion \(\widehat{X}\) is straight.
Reviewer: Kohzo Yamada (Shizuoka)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)Approximation and Baire classification of separately continuous functions on products of generalized ordered and compact spaces.https://zbmath.org/1460.260022021-06-15T18:09:00+00:00"Mykhaylyuk, Volodymyr"https://zbmath.org/authors/?q=ai:mykhaylyuk.volodymyr-vThe following two properties of a topological space \(X\) are examined.
\(X\) has Moran property (M), or a stronger approximation property (A), if for every compact space \(Y\) each separately continuous function \(f: X\times Y\to\mathbb{R}\) is of the first Baire class (in the case (M)), or there are continuous \(f_n:X\times Y\to\mathbb{R}\) such that \(f_n\restriction (\{x\}\times Y)\) and \(f_n\restriction (X\times \{y\})\) converge uniformly at every point to the corresponding restrictions of \(f\), respectively (in the case (A)).
The result from \textit{O. Karlova} and \textit{V. Mykhaylyuk} [Eur. J. Math. 3, No. 1, 87--110 (2017; Zbl 1370.26011)] says that a compact space \(X\) has property (M), or equivalently property (A), if and only if \(X\) has the countable chain condition with respect to cozero sets.
Here a necessary condition for the Moran property (M) is found, which coincides with perfectness (i.e., open sets are \(F_\sigma\)) for generalized order spaces. On the other hand, one of the main results says that generalized order spaces \(X\), which are simultaneously perfect, strongly zero-dimensional, hereditarily Baire, and have \(G_\delta\) diagonal in \(X\times X\), have the approximation property (A).
Using the existence of a Souslin continuum, a perfect, linearly ordered, and paracompact space \(X\) is found which does not have property (M). In fact, a compact space \(Y\) and a separately continuous \(f:X\times Y\to\mathbb{R}\) exist for that \(X\) such that \(f\) does not belong to \(\alpha\)'s Baire class for every \(\alpha<\omega_1\).
Concerning the methods, let us point out that among others several results on Namioka spaces are essentially used.
Reviewer: Petr Holický (Praha)On some properties of perfect images of GO-spaces and generalized trees.https://zbmath.org/1460.540192021-06-15T18:09:00+00:00"Peng, Liang-Xue"https://zbmath.org/authors/?q=ai:peng.liangxue"Wang, Huan"https://zbmath.org/authors/?q=ai:wang.huanIn this paper the authors study perfect images of generalized metric spaces, generalized ordered spaces and generalized trees. Let PIGO be the class of perfect images of generalized ordered spaces and PIGT be the class of perfect images of generalized trees with Hausdorff generalized Sorgenfrey topologies. It is an interesting question whether every monotonically normal paracompact space is a \(D\)-space, posed by \textit{C. R. Borges} and \textit{A. C. Wehrly} [Topol. Proc. 16, 7--15 (1991; Zbl 0787.54023)]. Every space in PIGO is monotonically normal. The following results are obtained in this paper: that every paracompact space in PIGO is a \(D\)-space; every \(k\)-semistratifiable space in PIGT is a metrizable space. The reviewer has the following question: Is a semi-stratifiable space in PIGT a developable space?
Reviewer: Shou Lin (Ningde)