## The Hausdorff fuzzy metric on compact sets.(English)Zbl 1069.54009

According to [A. George and P. Veeramani, Fuzzy Sets Syst. 64, 395–399 (1994; Zbl 0843.54014)], a fuzzy metric space is a triple $$(X,M,*)$$ where $$X$$ is a set, $$*$$ is a continuous $$t$$-norm and the fuzzy metric $$M$$ is a mapping $$M: X\times X\times (0,\infty) \to [0,1]$$ satisfying certain conditions. In the same paper the authors showed how this metric induces a topology $$\tau_M$$ on the set $$X$$.
The authors of this paper define a fuzzy metric $$H_M$$ on the set $$K(X)$$ of all nonempty compact subsets of $$(X,\tau_M)$$, that is actually a fuzzy set $$H_M: K(X)\times K(X)\times (0,\infty) \to [0,1]$$ which is a fuzzy analogue of the Hausdorff-Bourbaki metric, and study some properties of the corresponding space: such as completeness, completion and precompactness. A typical result states that the space is $$(X,M,*)$$ is precompact iff the space $$(K(X),H_M,*)$$ is precompact. Some examples are given. Possible applications of the obtained results are discussed.

### MSC:

 54A40 Fuzzy topology 54E35 Metric spaces, metrizability 54B20 Hyperspaces in general topology

Zbl 0843.54014
Full Text:

### References:

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