Fuzzy topology: Fuzzy closure operator, fuzzy compactness and fuzzy connectedness.

*(English)*Zbl 0809.54005By a fuzzy topology on a set \(X\) the authors realize a mapping \(\tau: I^ X\to I\) such that \(\tau(0)= \tau(1)=1\), \(\tau(U\wedge V)\geq \tau(U)\wedge \tau(V)\) for any \(U,V\in I^ X\), and \(\tau(\bigvee U_ i)\geq \bigwedge\tau(U_ i)\) for every family \(\{U_ i\): \(i\in {\mathcal I}\}\subset I^ X\). (The authors make reference to their paper [the authors with R. N. Hazra, ibid. 49, 237-242 (1992; Zbl 0762.54004)], however earlier such fuzzy topologies were considered in reviewer’s papers, see e.g. [Rend. Circ. Mat. Palermo, II. Ser. Suppl. 11, 89-103 (1985; Zbl 0638.54007)].)

In this paper a fuzzy topology is characterized by the so-called fuzzy closure operator which in fact is a mapping \(\text{Cl}: I^ X\times (0,1]\to I^ X\) satisfying certain conditions. Fuzzy closure operators are used, in particular, to characterize continuity of mappings of fuzzy topological spaces. The authors consider also the properties of compactness and connectedness for fuzzy topologies. These notions are introduced levelwise: namely, a fuzzy topology \(\tau\) is said to have a property \(P\) iff all its level Chang fuzzy topologies \(\tau^{-1} [\alpha,1]\), \(\alpha\in (0,1]\), have this property \(P\).

In this paper a fuzzy topology is characterized by the so-called fuzzy closure operator which in fact is a mapping \(\text{Cl}: I^ X\times (0,1]\to I^ X\) satisfying certain conditions. Fuzzy closure operators are used, in particular, to characterize continuity of mappings of fuzzy topological spaces. The authors consider also the properties of compactness and connectedness for fuzzy topologies. These notions are introduced levelwise: namely, a fuzzy topology \(\tau\) is said to have a property \(P\) iff all its level Chang fuzzy topologies \(\tau^{-1} [\alpha,1]\), \(\alpha\in (0,1]\), have this property \(P\).

Reviewer: A.Šostak (Riga)

##### MSC:

54A40 | Fuzzy topology |

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\textit{K. C. Chattopadhyay} and \textit{S. K. Samanta}, Fuzzy Sets Syst. 54, No. 2, 207--212 (1993; Zbl 0809.54005)

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##### References:

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