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  • Author or Editor: S. Romaguera x
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A bispace is called Dieudonn ´e complete if it admits a compatible bicomplete quasi- uniformity.We characterize pairwise Tychono .Dieudonn ´e complete bispaces in terms of their situation in the canonical bitopological compacti .cation and,from this result,derive a characterization of those T 0 topological spaces that admit a bicomplete quasi-uniformity. These characterizations are given in terms of bispaces which are 2perfect pre-images of bicompletely quasi-metrizable bispaces,which,in turn,are characterized in terms of the notion of an M -bispace.Using an example of R.Fox,we show that there exists a non quasi- metrizable bispace (X,P,Q)which has a Q ×P -G . -diagonal and is the 2perfect pre-image of a quasi-metrizable bispace,in contrast with the analogous metric topological theorem.

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Abstract  

V. Gregori and S. Romaguera [17] obtained an example of a fuzzy metric space (in the sense of A. George and P. Veeramani) that is not completable, i.e. it is not isometric to a dense subspace of any complete fuzzy metric space; therefore, and contrary to the classical case, there exist quiet fuzzy quasi-metric spaces that are not bicompletable neither D-completable, via (quasi-)isometries. In this paper we show that, nevertheless, it is possible to obtain solutions to the problem of completion of fuzzy quasi-metric spaces by using quasi-uniform isomorphisms instead of (quasi-)isometries. Such solutions are deduced from a general method, given here, to obtain extension properties of fuzzy quasi-metric spaces from the corresponding ones of the classical theory of quasi-uniform and quasi-metric spaces.

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A *-compactification of a T 1 quasi-uniform space (X,U ) is a compact T 1 quasi-uniform space (Y,V ) that has a T( V *)-dense subspace quasi-isomorphic to (X,U ), where V * denotes the coarsest uniformity finer than V .In this paper we characterize all Wallman type compactifications of a T 1 topological space in terms of the *-compactification of its point symmetric totally bounded transitive compatible quasi-uniformities. We deduce that the *-compactification of the Pervin quasi-uniformity of any normal T 1 topological space X is exactly the Stone-Cech compactification of X. We also obtain a characterization of those Hausdorff compactifications of a given space, which are of Wallman type.

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Let (X, d) be a quasi-metric space and (Y, q) be a quasi-normed linear space. We show that the normed cone of semi-Lipschitz functions from (X, d) to (Y, q) that vanish at a point x 0X, is balanced. Moreover, it is complete in the sense of D. Doitchinov whenever (Y, q) is a biBanach space.

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