\(\alpha\)-compactness in smooth topological spaces
Let \(X\) be a nonempty set and \(I = [0, 1]\). A smooth topology on \(X\) is a map \(\tau: I^X\longrightarrow I\) such that (1) \(\tau(0_X) = \tau(1_X) = 1\); (2) \(\tau(A\cap B)\geq\tau(A)\wedge\tau(B)\); (3) \(\tau(\cap_{j\in J}A_j) \geq \wedge_{j\in J}\tau(A_j)\). Let \(\tau\) be a smooth topology on \(X\). Then the map \(\tau^*: I^X\longrightarrow I\) defined by \(\tau^*(A) = \tau(1-A)\) is a smooth cotopology (which is defined similarly) on \(X\). For each \(\alpha\in[0, 1]\), the family \(\tau_{\alpha} = \{A\in I^X : \tau(A) \geq \alpha\}\) is a Chang fuzzy topology on \(X\). The family of closed sets with respect to \(\tau_{\alpha}\) is given by \(\tau^*_{\alpha} = \{A\in I^X : \tau^*(A) \geq \alpha\}\). The authors define the \(\alpha\)-closure and \(\alpha\)-interior in a smooth topological space as follows: for all \(A\in I^X\), the \(\alpha\)-closure of \(A\), denoted \(\overline{A}_{\alpha}\), is defined to be \(\cap\{K\in I^X : \tau^*(K)>\alpha\tau^*(A), A\subseteq K\}\); the \(\alpha\)-interior of \(A\) is defined similarly. It should be noted that the \(\alpha\)-closure is not the closure with respect to the \(\alpha\)-level topology \(\tau_{\alpha}\). Some properties of \(\alpha\)-closure and \(\alpha\)-interior are discussed in this paper. The \(\alpha\)-closure and \(\alpha\)-interior are employed to introduce several notions of compactness for smooth topological spaces in a canonical way.
scientific article; zbMATH DE number 2012584