Some other types of fuzzy connectedness (Q1187437)

Various concepts of fuzzy connectedness are investigated. \(\alpha\)-level connectedness concepts \(\alpha\)-\(C3\) and \(\alpha\)-\(C4\) are introduced and studied. Some of the theorems are as follows. The following are equivalent: (1) \((X,t)\) is \(\alpha\)-\(C4\); (2) \((X,i_{1-\alpha}(t))\) is connected. \(\alpha\)-\(C3\), \(\alpha\)-\(C4\) and \(S\)-\(C4\) are preserved under continuous functions. \(\alpha\)-\(C3\), \(\alpha\)-\(C4\), \(S\)-\(C4\) and \(\alpha\)-\(C\) are good extensions. A non-empty product space is \(\alpha\)-\(C3\) (resp. \(\alpha\)-\(C4\) or \(S\)-\(C4\)) iff each factor space is \(\alpha\)-\(C3\) (resp. \(\alpha\)-\(C4\) or \(S\)-\(C4\)). Various concepts of fuzzy connectedness are compared. Many examples are given.