zbMATH — the first resource for mathematics

On fuzzy uniform spaces. (English) Zbl 0598.54003
We give some results about fuzzy uniform spaces. Our definition of a fuzzy uniform space is that of Hutton with the only difference that every member \(\alpha\) of a fuzzy uniformity in our sense is such that \(\alpha (0)=0\). This is in accordance with what happens in the ordinary uniform spaces. The notion of a fuzzy uniform space given by Lowen differs from our concept of a fuzzy uniform space. We show that to every uniformity \({\mathcal U}\) on a set X corresponds a fuzzy uniformity \(\phi\) (\({\mathcal U})\) and that to every fuzzy uniformity \(\Phi\) on X corresponds a uniformity \(\psi\) (\(\Phi)\). The fuzzy topology generated by a uniformizable topology is uniformizable. In the last section we prove that for every fuzzy proximity \(\delta\), the class \(\Pi\) (\(\delta)\) of all fuzzy uniformities which are compatible with \(\delta\) is not empty and that \(\Pi\) (\(\delta)\) contans a smallest member \({\mathcal U}(\delta)\).

54A40 Fuzzy topology
54E15 Uniform structures and generalizations
54E05 Proximity structures and generalizations
Full Text: DOI
[1] Chang, C.L, Fuzzy topological spaces, J. math. anal. appl., 24, 182-190, (1968) · Zbl 0167.51001
[2] Császár, A, Foundations of general topology, (1963), Pergamon Press London
[3] Erceg, M.A, Metric spaces in fuzzy set theory, J. math. anal. appl., 69, 205-230, (1979) · Zbl 0409.54007
[4] Höhle, U, Probabilistic uniformization of fuzzy topologies, Fuzzy sets and systems, 1, 311-332, (1978) · Zbl 0413.54002
[5] Hutton, B.W, Uniformities on fuzzy topological spaces, J. math. anal. appl., 58, 559-571, (1977) · Zbl 0358.54008
[6] Hutton, B.W, Products of fuzzy topological spaces, Topology and appl., 11, 59-67, (1980) · Zbl 0422.54006
[7] Katsaras, A.K, Fuzzy proximity spaces, J. math. anal. appl., 68, 100-110, (1979) · Zbl 0412.54006
[8] Katsaras, A.K, On fuzzy proximity spaces, J. math. anal. appl., 75, 571-583, (1980) · Zbl 0443.54006
[9] Katsaras, A.K, Fuzzy proximities and fuzzy completely regular spaces, I, Anal. şt. univ. iaşi, 26, 31-41, (1980) · Zbl 0443.54007
[10] Katsaras, A.K, Fuzzy topological vector spaces, I, Fuzzy sets and systems, 6, 85-95, (1981) · Zbl 0463.46009
[11] Katsaras, A.K; Petalas, C.G, A unified theory of fuzzy topologies, fuzzy proximities and fuzzy uniformities, Rev. roumaine math. pures appl., 28, 9, 845-856, (1983) · Zbl 0521.54002
[12] Lowen, R, Fuzzy uniform spaces, J. math. anal. appl., 82, 370-385, (1981) · Zbl 0494.54005
[13] Petalas, C.G, Fuzzy syntopogenous structures, () · Zbl 1070.54004
[14] Rodabaugh, S, The Hausdorff separation axiom for fuzzy topological spaces, Topology and appl., 11, 319-334, (1980) · Zbl 0484.54008
[15] Srivastava, P; Gupta, R.L, Fuzzy proximity bases and subbases, J. math. anal. appl., 78, 588-597, (1980) · Zbl 0481.54006
[16] Warren, R.H, Neighborhoods bases and continuity in fuzzy topological spaces, Rocky mountain J. math., 8, No. 3, 459-470, (1978) · Zbl 0394.54003
[17] Zadeh, L.A, Fuzzy sets, Inform. contr., 8, 338-353, (1965) · Zbl 0139.24606
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.