\(T\)-product fuzzy topology (Q1313027)

The \(T\)-product \(\tau_ 1 \otimes_ T \tau_ 2\) on \(X_ 1 \times X_ 2\) of fuzzy topological spaces \((X_ 1, \tau_ 1)\) and \((X_ 2, \tau_ 2)\) is defined. Some properties of the \(T\)-product are proved. The projection mappings \(p_ i : {(X_ 1 \times X_ 2, \tau_ 1 \otimes_ T \tau_ 2)}\) \(\to (X_ i, \tau_ i)\), \(i = 1,2\), are continuous. If \((X_ i, \tau_ i)\) are fuzzy Hausdorff spaces then the \(T\)-product is fuzzy Hausdorff, too. If \((X_ 1, \tau_ 1)\) and \((X_ 2, \tau_ 2)\) are separable then the \(T\)-product is also separable. \((X_ 1 \times X_ 2, \tau_ 1 \otimes_ T \tau_ 2)\) is compact iff \((X_ i, \tau_ i)\), \(i = 1,2\), are compact.