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**Generalizing the L-fuzzy unit interval.**
*(English)*
Zbl 0549.54004

The L-fuzzy unit interval, which was introduced by B. Hutton [J. Math. Anal. Appl. 50, 74-79 (1975; Zbl 0297.54003)], and the L-fuzzy real line play a role in fuzzy topology analogous to the role of the unit interval and the reals in topology. In addition, they are among the most difficult examples of fuzzy spaces commonly encountered at present. [For a survey of the L-fuzzy real line and its subspaces, see S. E. Rodabaugh, Recent developments in fuzzy set and possibility theory, R. R. Yager (ed.) (Pergamon Press 1982).]

The construction of these important spaces at first appears to be unavoidably dependent on the order properties of the reals. This paper presents a generalization of this construction which applies to any connected topological space. If L is linearly ordered, the space associated with an interval of real numbers is fuzzy homeomorphic in a simple way to the fuzzy space constructed by the standard method. For many spaces, including \(R^ n\) for all n, there is a simple embedding of the original space in the associated L-fuzzy space. Finally, when X is compact and L is linearly ordered, the space of monotone maps is a subspace of the constructed space X(L) and so can be endowed with the subspace fuzzy topology.

The construction of these important spaces at first appears to be unavoidably dependent on the order properties of the reals. This paper presents a generalization of this construction which applies to any connected topological space. If L is linearly ordered, the space associated with an interval of real numbers is fuzzy homeomorphic in a simple way to the fuzzy space constructed by the standard method. For many spaces, including \(R^ n\) for all n, there is a simple embedding of the original space in the associated L-fuzzy space. Finally, when X is compact and L is linearly ordered, the space of monotone maps is a subspace of the constructed space X(L) and so can be endowed with the subspace fuzzy topology.

### MSC:

54A40 | Fuzzy topology |

26A99 | Functions of one variable |

54B99 | Basic constructions in general topology |

### Keywords:

L-fuzzy unit interval; L-fuzzy real line; fuzzy topology; fuzzy homeomorphic; L-fuzzy space; space of monotone maps; subspace fuzzy topology### Citations:

Zbl 0297.54003
Full Text:
DOI

### References:

[1] | Birkhoff, G, Lattice theory, () · Zbl 0126.03801 |

[2] | Hutton, B, Normality in fuzzy topological spaces, J. math. anal. appl., 50, 74-79, (1975) · Zbl 0297.54003 |

[3] | Klein, A.J, Α-closure in fuzzy topology, Rocky mtn. J. math., 11, 553-560, (1981) · Zbl 0484.54009 |

[4] | Klein, A.J, Generating fuzzy topologies with semi-closure operators, Fuzzy sets and systems, 9, 267-274, (1983) · Zbl 0518.54009 |

[5] | Lowen, R, Fuzzy topological spaces and fuzzy compactness, J. math. anal. appl., 56, 621-633, (1976) · Zbl 0342.54003 |

[6] | Rodabaugh, S.E, The L-fuzzy real line and its subspaces, () · Zbl 0508.54002 |

[7] | Warren, R.H, Neighborhoods, bases and continuity in fuzzy topological spaces, Rocky mtn. J. math., 8, 459-470, (1978) · Zbl 0394.54003 |

[8] | Whyburn, G.T, Dynamic topology, Amer. math. monthly, 77, 556-570, (1970) · Zbl 0203.55502 |

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