Let \((X,\delta)\) be a fuzzy topological space. If \(S(X)\) is the Boolean algebra of all subsets of \(X\) then to every fuzzy set \(f\) on \(X\) a fuzzy set \(f^\diamondsuit\) on \(S(X)\) is defined as follows: \[ f^\diamondsuit(A)= \sup\{f(X)\mid x\in A\}, \qquad A\in S(X). \] Let the family \(\{f^\diamondsuit\mid f\in\delta\}\) be a subbase for a fuzzy topology \(\delta^\diamondsuit\) on \(S(X)\). The relation between a fuzzy topological space \((X,\delta)\) and the fuzzy topological Boolean algebra \((S(X),\delta^\diamondsuit)\) is investigated. If \({\mathcal B}\) is a Boolean algebra, \({\mathcal J}\) an ideal of \({\mathcal B}\) and \(\sigma\) a fuzzy topology on \({\mathcal B}\), then two quotient topologies \(k_1(\sigma)\) and \(k_2(\sigma)\) on \({\mathcal B}/{\mathcal J}\) are defined. Some results are as follows.
A fuzzy topological space is \(\text{FT}_i\) iff \((S(X), \delta^\diamondsuit)\) is \(\text{FT}_i\), \(i=1,2\). If \((S(X),\sigma)\) is a fuzzy topological Boolean algebra then \(\sigma_X= \{(f)_X\mid f\in\sigma\}\) is a fuzzy topology on \(X\) having the following property: \((X,\sigma_X)\) is \(\text{FT}_i\iff (S(X),\sigma)\) is \(\text{FT}_i\), \(i=1,2\). Some properties of continuity and of quotient fuzzy topological Boolean algebra are proved.