The role of Sierpiński object in fuzzy topology (Q2754360)

This is an interesting survey of the role of the fuzzy Sierpiński spaces in the category of fuzzy topological spaces and that of fuzzy bitopological spaces. A fuzzy topology \(\Delta\) on a set \(X\) is a subset of \([0, 1]^X = I^X\) which is closed with respect to finite infs and arbitrary sups. \(\Delta\) is said to be stratified if it contains all the constant functions \(X\rightarrow I\). The category of fuzzy topological spaces is denoted \textbf{FTS} and the full subcategory of \textbf{FTS} consisting of stratified fuzzy topological spaces is denoted \textbf{L-FTS}. Then \textbf{FTS} is a topological category and \textbf{L-FTS} is a well-fibred topological category. NEWLINENEWLINENEWLINELet \(I_S = (I, S)\), where \(I = [0, 1]\) and \(S = \{0, id, 1\}\). Then \(I_S\) is a fuzzy topological space, and it is a Sierpiński object for the category \textbf{FTS} in the sense of \textit{E. G. Manes} [Algebraic theories, Grad. Texts Math. 26 (1976; Zbl 0353.18007)]. So \(I_S\) is called the fuzzy Sierpiński space. Similarly, let \(I_{S^c} = (I, S^c)\), where \(I = [0, 1]\) and \(S^c = \{(\alpha\wedge id)\vee\beta\}\). Then \(I_{S^c}\) is a stratified fuzzy topological space, and it is a Sierpiński object for the category \textbf{L-FTS}.NEWLINENEWLINENEWLINEIn this paper, properties of the fuzzy Sierpiński space, most of which were obtained by the author and his collaborators, are surveyed. These results demonstrate that the fuzzy Sierpiński space \(I_S\) `behaves' almost in the same way as its topological counterpart, the Sierpiński space \(2_S\), does. Typical results are: (1) The subcategory \textbf{FTS}\(_0\) (\textbf{L-FTS}\(_0\)) of \(T_0\) (stratified) fuzzy topological spaces is the epireflective hull of \(I_S\) (\(I_{S^c}\)) in \textbf{FTS} (\textbf{L-FTS}). (2) An object in \textbf{FTS}\(_0\) is injective iff it is a retract of some powers of \(I_S\). (3) The category of fuzzy sober spaces (in the sense of Rodabaugh) is the epireflective hull of \(I_S\) in \textbf{FTS}\(_0\). NEWLINENEWLINENEWLINEIn the last three sections, results about Sierpiński objects in the category of fuzzy bitopological spaces and their relations to other concepts, such as sobriety, etc, are surveyed.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00008].