On hereditary precompactness and completeness in quasi-uniform spaces.(English)Zbl 0924.54035

The author studies left (and right) $$K$$-complete quasi-uniform spaces. A quasi-uniformity on a set $$X$$ is a filter $${\mathcal U}$$ on $$X\times X$$ such that for each $$U\in{\mathcal U}$$, (i) $$\{(x,x):x\in X\}\subseteq U$$ and (ii) $$V^2\subseteq U$$ for some $$V\in{\mathcal U}$$. A quasi-uniformity $${\mathcal U}$$ on a set $$X$$ determines the topology $$T({\mathcal U})$$ on $$X$$ in the standard way. Let $$X=(X,{\mathcal U})$$ be a quasi-uniform space. Then, $$X$$ is called uniformly regular if for each $$U\in{\mathcal U}$$ there is $$V\in{\mathcal U}$$ such that $$\operatorname{cl}_{T({\mathcal U})}V(x)\subseteq U(x)$$ for each $$x\in X$$. A filter $${\mathcal F}$$ on $$X$$ is called left (resp. right) $$K$$-Cauchy if for each $$U\in{\mathcal U}$$ there is $$F\in{\mathcal F}$$ such that $$U(x)\in{\mathcal F}$$ (resp. $$U^{-1}(x)\in{\mathcal F}$$) for all $$x\in F$$. Further, $$X$$ is called left (resp. right) $$K$$-complete if every left (resp. right) $$K$$-Cauchy filter converges in the topology $$T({\mathcal U})$$. Some of the main results are:
(1) A $$T_0$$ uniformly regular quasi-uniform space is left $$K$$-complete if and only if it is Smyth-complete in the sense of M. B. Smyth [J. Lond. Math. Soc., II. Ser. 49, No. 2, 385-400 (1994; Zbl 0798.54036)];
(2) if a uniformly regular quasi-uniform space $$(X,{\mathcal U})$$ is left $$K$$-complete, then $$(X,{\mathcal U}^{-1})$$ is right $$K$$-complete, where $${\mathcal U}^{-1}=\{U^{-1}:U\in{\mathcal U}\}$$.

MSC:

 5.4e+16 Uniform structures and generalizations

Zbl 0798.54036
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References:

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