According to A. George and P. Veeramani [Fuzzy Sets Syst. 64, No. 3, 395–399 (1994; Zbl 0843.54014)], a fuzzy metric space is a triple \((X,M,\ast)\), where \(X\) is a set, \(\ast\) is a continuous \(t\)-norm [B. Schweizer and A. Sklar, Probabilistic Metric Spaces. North Holland Series in Probability and Applied Mathematics. New York-Amsterdam-Oxford: North-Holland. (1983; Zbl 0546.60010)], and \(M: X\times X\times(0,\infty)\rightarrow[0,1]\) is a map (fuzzy metric on \(X\)), which satisfies the following conditions for every \(x\), \(y\), \(z\in X\) and every \(s\), \(t>0\): (1) \(M(x,y,t)>0\); (2) \(M(x,y,t)=1\) if and only if \(x=y\); (3) \(M(x,y,t)=M(y,x,t)\); (4) \(M(x,y,t)\ast M(y,z,s)\leqslant M(x,z,t+s)\); (5) the map \(M(x,y,-): (0,\infty)\rightarrow[0,1]\) is continuous.
The current paper considers a fuzzy metric space analogue of the result that given a point \(x_0\) in a (crisp) metric space \((X,d)\), and a family \(\xi\) of open balls centered at \(x_0\), if the intersection of the elements of \(\xi\) is the singleton set \(\{x_0\}\), then \(\xi\) is a local base at \(x_0\), provided that \(x_0\) is not isolated in \((X,d)\).
The authors begin with an easy example (Example 7 on page 144), which shows that the above-mentioned claim is no longer true in fuzzy metric spaces. When restricting oneself, however, to stationary fuzzy metric spaces (which means that for every \(x\), \(y\in X\), the map \(M_{x,y}(t)=M(x,y,t)\) is constant), one does get the required statement for particular families of open balls (Proposition 8 on page 145). Similar analogues are obtained for principal fuzzy metric spaces (Corollary 14 on page 146), and their dual co-principal fuzzy metric spaces (Corollary 22 on page 147), as well as replacing “co-principal” with “compact” (Theorem 23 on page 147). The authors though were unable to present a general analogue of the above-mentioned claim in case of fuzzy metric spaces, formulating instead an open problem (Problem 17 on page 146).
The paper is well written, and is easy to follow, provided that the reader possesses a certain background on the theory of fuzzy metric spaces (which can be obtained from the numerous references given by the authors on this topic).