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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.)

Citations:

Zbl 0649.54015
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