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Nearness-based topology. (English) Zbl 0993.54006
A binary fuzzy relation $$N$$ on a universe $$X$$ is called a nearness on $$X$$ if $$N(x,x)=1 (x\in X)$$, $$N(x,y)=N(y,x) (x,y \in X)$$, and for every $$\varepsilon >0$$ there exists $$\delta <1$$ such that $$N(x,y)>\delta$$ implies $$|N(x,z)- N(y,z) |< \varepsilon (x,y,z \in X).$$ The corresponding notions of $$N$$-topology, $$N$$-convergence, $$N$$-continuity are investigated, and a fixed point theorem on $$N$$-topological space is established.

##### MSC:
 54A40 Fuzzy topology 54E17 Nearness spaces
##### Keywords:
nearness; $$N$$-topology