Fuzzy topological properties of fuzzy points and its applications (Q2839862)

Motivated by the concept of fuzzy point of~\textit{P.-M.~Pu} and \textit{Y.-M.~Liu} [J. Math. Anal. Appl. 76, 571--599 (1980; Zbl 0447.54006)] and the result of~\textit{D.~S.~Janković} and \textit{I.~L.~Reilly} [Indian J. Pure Appl. Math. 16, 957--964 (1985; Zbl 0572.54010)] (which claims that given an element \(x\) of a topological space \(X\), it follows that either \(x\in Int(Cl(\{x\}))\) (\(\{x\}\) is pre-open) or \(Int(Cl(\{x\}))=\varnothing\) (\(\{x\}\) is nowhere dense)), the manuscript under review shows that the above claim of D.~S.~~Janković and I.~L.~Reilly is no longer true in fuzzy topological spaces of~\textit{C.~L.~Chang} [J. Math. Anal. Appl. 24, 182--190 (1968; Zbl 0167.51001)] (Remark~2.2 on page~238). More particular, the authors show that given a fuzzy topological space \(X\), every fuzzy set \(\lambda\in I^X\backslash\{0_X\}\) (where \(I\) is the unit interval \([0,1]\)) has the form of \(\lambda_{\mathcal{FPO}}\vee\lambda_{\mathcal{FND}}\vee\lambda_{\mathcal{REST}}\), where \(\lambda_{\mathcal{FPO}}\) (resp. \(\lambda_{\mathcal{FND}}\)) is the join of all those fuzzy points of \(\lambda\), which are fuzzy pre-open (resp. fuzzy nowhere dense) sets, and \(\lambda_{\mathcal{REST}}\) is the join of all the other fuzzy points. Moreover, the above three fuzzy components of \(\lambda\) do not intersect, which means, e.g., \(\lambda_{\mathcal{FPO}}\wedge\lambda_{\mathcal{FND}}=0_X\) (Theorem~2.10 on page~241). The paper then shows several examples (some of which positioned as applications) of such decompositions (Section~3 on pages~242~-~251), including, e.g., the digital plane of~\textit{E.~Khalimsky, R.~Kopperman} and \textit{P.~R.~Meyer} [Topology Appl. 36, No.~1, 1--17 (1990; Zbl 0709.54017)], in which one considers the fuzzy topology, whose elements are precisely the characteristic maps of the elements of the crisp topology in question (pages~248~-~250).NEWLINENEWLINEThe paper is rather badly written (a lot of typos), its presented results are quite straightforward and easy, and the claimed applications seem a bit obscure.