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**Fuzzy topological vector spaces. II.**
*(English)*
Zbl 0555.46006

This is a continuation of ibid. 6, 85-95 (1981; Zbl 0463.46009). It is shown that a topology \(\tau\), on a vector space E, is linear iff the fuzzy topology \(\omega\) (\(\tau)\), consisting of all \(\tau\)-lower semicontinuous fuzzy sets, is linear. The fuzzy seminormed and the fuzzy normed linear spaces are introduced and some of their properties are studied. It is also given the concept of a bounded fuzzy set and the concept of a bornological fuzzy linear space. A linear topology \(\tau\) on E is bornological iff \(\omega\) (\(\tau)\) is bornological. The locally convex fuzzy linear topologies form a special class of fuzzy linear topologies. Some of the properties of these spaces as well as of the quotient spaces and the bornological spaces, are investigated.

### MSC:

46A99 | Topological linear spaces and related structures |

54A40 | Fuzzy topology |

46A08 | Barrelled spaces, bornological spaces |

46A03 | General theory of locally convex spaces |

### Keywords:

fuzzy topology; fuzzy normed linear spaces; bounded fuzzy set; bornological fuzzy linear space; locally convex fuzzy linear topologies### Citations:

Zbl 0463.46009
Full Text:
DOI

### References:

[1] | Chang, C. L., Fuzzy topological spaces, J. Math. Anal. Appl, 24, 182-190 (1968) · Zbl 0167.51001 |

[2] | Katsaras, A. K.; Liu, D. B., Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl, 58, 135-146 (1977) · Zbl 0358.46011 |

[3] | Katsaras, A. K., Fuzzy topological vector spaces I, Fuzzy Sets and Systems, 6, 85-95 (1981) · Zbl 0463.46009 |

[4] | Lowen, R., Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl, 56, 621-633 (1976) · Zbl 0342.54003 |

[5] | Lowen, R., Convex fuzzy sets, Fuzzy Sets and Systems, 3, 291-310 (1980) · Zbl 0439.52001 |

[6] | Pao-Ming, Pu; Ying-Ming, Liu, Fuzzy Topology I. Neighborhood structure of a point and Moore-Smith convergence, J. Math. Anal. Appl, 76, 571-599 (1980) · Zbl 0447.54006 |

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

[8] | Warren, R. H., Fuzzy topologies characterized by neighborhood systems, Rocky Mountain J. Math, 9, 761-764 (1979) · Zbl 0429.54003 |

[9] | Wong, C. K., Fuzzy topology: product and quotient theorems, J. Math. Anal. Appl, 45, 512-521 (1974) · Zbl 0273.54002 |

[10] | Zadeh, L. A., Fuzzy Sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606 |

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