## A new class of quasi-uniform spaces.(English)Zbl 0949.54032

Recall that for filters $$\mathcal F$$ and $$\mathcal D$$ on a quasi-uniform space $$(X,\mathcal U)$$ the notation $$(\mathcal D,\mathcal F)\to 0$$ means that for each $$U\in \mathcal U$$ there are a $$G\in \mathcal D$$ and an $$F\in \mathcal F$$ such that $$G\times F\subset U$$. The author calls a quasi-uniform space $$(X,\mathcal U)$$ fitting if for each $$U\in \mathcal U$$ there exists a $$V\in \mathcal U$$ such that whenever $$\mathcal F$$ and $$\mathcal D$$ are filters on $$X$$ with $$(\mathcal D,\mathcal F)\to 0$$ and $$x$$ and $$y$$ are points of $$X$$ satisfying $$(V\cap V^{-1})(x)\in\mathcal F$$ and $$(V\cap V^{-1})(y)\in \mathcal D$$, then $$(x,y)\in U$$. Every quiet quasi-uniform space in the sense of D. Doichinov [C. R. Acad. Bulg. Sci. 41, No. 7, 5-8 (1988; Zbl 0649.54015)] is fitting. It is shown that every bicompletion of a fitting quasi-uniform space is again a quasi-uniform space. Moreover, generalizing a result of Fletcher and Hunsaker from quiet quasi-uniform spaces to fitting quasi-uniform spaces, it is shown that every totally bounded fitting quasi-uniform space is a uniform space. Every fitting quasi-uniform space $$(X,\mathcal U)$$ is uniformly $$R_0$$, i.e. it has the property that for each $$U\in \mathcal U$$ there is a $$V\in \mathcal U$$ such that $$\text{cl}_{\tau_{(\mu)}}(V\cap V^{-1})(x)\subset U(x)$$ for each $$x\in X$$. It is shown that for uniformly $$R_0$$ Smyth completable quasi-uniform spaces four concepts of completeness for quasi-uniform spaces coincide. Additionally, $$T_1$$ quasi-uniform spaces which have a $$T_1$$ quasi-uniform bicompletion are characterized.

### MSC:

 54E15 Uniform structures and generalizations 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)

Zbl 0649.54015
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