Quasi-uniform Spaces Research Articles

D-Complete extensions of quasi-uniform spaces

D-Complete extensions of quasi-uniform spaces

CHARACTERIZATIONS OF FRAME QUASI-UNIFORMITIES

Abstract This paper advances an entourage-like theory of quasi-uniformities for frames by establishing that the theory comprises the classical theory of quasi-uniform spaces and is, in the setting of biframes, equivalent to J.L. Frith's theory of covering quasi-uniformities.

Stability in quasi-uniform spaces and the inverse problem

We study the concept of a stable quasi-uniform space due to D. Doitchivov. In particular quasi-pseudo-metric spaces ( X,d) whose associated quasi-uniformities U ( d) or U ( d -1) are stable are investigated.

Symmetry conditions in terms of open sets

A quasi-uniform space is Smyth symmetric provided that its quasi-proximity is a proximity. This paper shows that a space is Smyth symmetric if and only if it is small-set symmetric and either open symmetric or semi-symmetric in the sense of J. Deák. Every Smyth symmetric quasi-uniformity is quiet, and in the class of totally bounded quasi-uniform spaces both Smyth symmetry and quietness are equivalent to being a uniformity. In the class of small-set symmetric spaces the concepts of open symmetry, semi-symmetry and Smyth symmetry are equivalent.

Completeness using pairs of filters

This paper considers three kinds of completeness: D-completeness, strong D-completeness, and pair completeness. Although these three concepts can be defined for arbitrary quasi-uniform spaces, their relationships are particularly important in the class of uniformly regular quasi-uniform spaces, a class of spaces which includes all quiet spaces. We show that in a uniformly regular space D-completeness implies pair completeness; and, although pair completeness does not imply D-completeness even in the class of quiet spaces, we show that every equinormal regular pair complete space is D-complete. As the terminology suggests, every strongly D-complete space is D-complete. We show that a uniformly regular space is strongly D-complete if and only if it is D-complete and point symmetric. D-completeness and strong D-completeness also coincide in the class of locally symmetric spaces, but we give an example to show that D-completeness and pair completeness do not coincide in this class. It follows from our results that the three concepts of completeness agree in a uniform space or in any equinormal uniformly regular space.

Fuzzy quasi uniformities and fuzzy bitopological spaces

The notions of a fuzzy bitopological space and some separation axioms are introduced. Also, the relation between a fuzzy bitopological space and a fuzzy quasi uniform space is investigated. As a main result, there is given a theorem: a fuzzy bitopological space is fuzzy quasi uniformizable iff it is pairwise completely regular. Moreover, if the space is pairwise compact, then the fuzzy bitopology can be induced by a unique fuzzy quasi uniformity.

Functorial admissible quasi-uniformities on topological spaces

Let T denote the forgetful functor from the category Quu of quasi-uniform spaces and quasi-uniformly continuous maps to the category Top of topological spaces and continuous maps. In the first part of this paper we develop a general method to construct nontransitive admissible quasi-uniformities on topological spaces. We use this technique to answer a question of G.C.L. Brümmer in the negative by exhibiting a nontransitive T-section that is coarser than the (transitive) locally finite covering quasi-uniformity. In the second part of this paper we show that the Pervin functor restricted to the category Haus of Hausdorff spaces and continuous maps is the coarsest functor that puts compatible quasi-uniformities on the Hausdorff spaces.

Entourage uniformities for frames

The purpose of this paper is to define a uniformity on a frame using entourages. A definition of uniformities in terms of covers has been given byA. Pultr ([8], [10]). We introduce two new symmetry conditions for a quasi-uniform space, open-set symmetry and small-set symmetry. We prove that a quasi-uniformityU is a uniformity if and only if it is both open-set symmetric and small-set symmetric. The category of (covering) uniform frames is isomorphic with the category of entourage uniform frames.

A concept of completeness of quasi-uniform spaces

We define a notion of completion for quasi-uniform spaces in a categorical manner, and construct such completions for a large class of spaces (called quiet quasi-uniform spaces). In the uniform case, this completion coincides with the usual completion.

Uniformly regular quasi-uniformities

The concept of uniform regularity is studied. Every quiet quasi-uniform space is uniformly regular, and every point symmetric uniformly regular quasi-uniform space that is complete (in the sense of Doitchinov) is quiet. Every Lebesgue quasi-uniformity (in particular every compact quasi-uniformity) that is uniformly regular is quiet. Every continuous quasi-metric is uniformly regular, but the Michael line has a continuous quasi-metric that is not quiet.

Some remarks on quasi-uniform spaces

A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.

Sobrification and bicompletion of totally bounded quasi-uniform spaces

AbstractWe observe that if is a compatible totally bounded quasi-uniformity on a T0-space (X,), then the bicompletion of (X, ) is a strongly sober, locally quasicompact space. It follows that the b-closure S of (X, ) in is homeomorphic to the sobrification of the space (X, ). We prove that S is equal to if and only if (X, ) is a core-compact space in which every ultrafilter has an irreducible convergence set and is the coarsest quasi-uniformity compatible with . If is the Pervin quasi-uniformity on X, then S is equal to if and only if X is hereditarily quasicompact, or equivalently, is the Pervin quasi-uniformity on .

Compactifications of totally bounded quasi-uniform spaces

The notation and terminology of this paper coincide with that of reference [4], except that here the term, compactification, refers to a T1-space. It is known that a completely regular totally bounded Hausdorff quasi-uniform space (X, ) has a Hausdorff compactification if and only if contains a uniformity compatible with ℱ() [4, Theorem 3.47]. The use of regular filters by E. M. Alfsen and J. E. Fenstad [1] and O. Njåstad [5], suggests a construction of a compactification, which differs markedly from the construction obtained in [4]. We use this construction to show that a totally bounded T1 quasi-uniform space has a compactification if and only if it is point symmetric. While it is pleasant to have a characterization that obtains for all T1-spaces, the present construction has several further attributes. Unlike the compactification obtained in [4], the compactification given here preserves both total boundedness and uniform weight, and coincides with the uniform completion when the quasi-uniformity under consideration is a uniformity. Moreover, any quasi-uniformly continuous map from the underlying quasi-uniform space of the compactification onto any totally bounded compact T1-space has a quasi-uniformly continuous extension to the compactification. If is the Pervin quasi-uniformity of a T1-space X, the compactification we obtain is the Wallman compactification of (X, ℱ ()). It follows that our construction need not provide a Hausdorff compactification, even when such a compactification exists; but we obtain a sufficient condition in order that our compactification be a Hausdorff space and note that this condition is satisfied by all uniform spaces and all normal equinormal quasi-uniform spaces. Finally, we note that our construction is reminiscent of the completion obtained by Á. Császár for an arbitrary quasi-uniform space [2, Section 3]; in particular our Theorem 3.7 is comparable with the result of [2, Theorem 3.5].

Nearness and quasi-uniform structures

The concept of a locally right symmetric quasi-uniform space is introduced. Every symmetric topological space admits a locally right symmetric quasi-uniform structure. It is shown that this property characterizes those quasi-uniform spaces for which the collection of quasi-uniform covers forms a nearness structure with the same closure operator. With this link, various concepts and results in these two areas are compared.

Uniformization of quasi-uniform spaces

This paper considers the question of when a quasi-uniform space has a compatible uniform structure. Typical of the sufficient conditions provided is the result that a quasi-uniform space whose conjugate topology is compact and R0 is uniformizable.

On box products of syntopogenous spaces

The syntopogenous structure is a common generalization of topology, proximity and quasi-uniformity. Actually, the category Snt of syntopogenous spaces contains the category Top of topological spaces, the category Prox of proximity spaces and the category q-Unif of quasi-uniform spaces as bicoreflective subcategories. The category Snt has product which induces the product in Top and in q-Unif, of course. The aim of this paper is to define and investigate another kind of product which gives the box product in Top and in q-Unif. We are grateful to Professor A. Cs~sz~r for some helpful comments and for saving us from several errors. Our basic references are [4] on box products and the monograph [2] on syntopogenous spaces. For convenience we recall the main definitions. A binary relation -< defined on the subsets of a set X is said to be a topogenous order provided that the following conditions are satisfied: (01) O < ~ , X < X , (02) A < B implies A c B , (03) A c A ' < B ' c B implies A<B, (Q) A<B and A'<B" imply AUA'<BUB" and Af~A'<BAB'. A pair (X, S) consisting of a set X and a nonempty family S of topogenous orders on X is called syntopogenous space provided the following axioms hold: (S1) for 1<, 2<E S there is a z<ES with the property 1< U 2 < c 3 < , ($2) for I < E S there is a z<ES with the property ~ < o 2 < D < 1. If S and S' are syntopogenous structures on the set X then S and S" are equivalent provided that for any < E S there is a < ' E S ' such that < c < ' and for any < ' E S ' there is a < E S such that < ' c < . A syntopogenous space (X, S) is topological if S = {<} and < satisfies the following axiom:

A Construction of the Pair Completion of a Quasi-Uniform Space

Although L. Nachbin introduced the concept of a quasiuniform space in connection with the study of uniform ordered spaces [7], it was A. Császár who first developed a theory of completion for these spaces. In the review of Császár′s work [3], J. Isbell wrote: “A topogenic space is already complete according to the present definition; an unfortunate side effect is that not every convergent filter is Cauchy.” This aspect of Császár′s theory of completeness seems to have stalled the investigation of its applications, so that even in the study of uniform ordered spaces Császár′s important work has often been overlooked. While it may have seemed that Isbell had pointed out the Achilles′ heel of Császár′s theory, in this paper we outline a construction of Császár′s completion for quasi-uniform spaces, which differs from Császár′s original construction and the construction given by S. Salbany [8], in which Isbell′s objection does not obtain.

On Precompact (Quasi-) Uniform Structures

The following are shown: (1) The subbasis theorem on precompactness does not hold even in uniform spaces. (2) The supremum of (even finitely many) precompact quasi-uniform structures is not necessarily precompact. (3) A compact quasi-uniform space is not necessarily totally bounded. These results contradict corresponding assertions in the literature.

ON CERTAIN FACTORIZATIONS OF FUNCTORS INTO THE CATEGORY OF QUASI-UNIFORM SPACES

This paper is motivated by the search for natural extensions of classical uniform space results to quasi-uniform spaces. As instances of such extensions we restate some theorems of P. Fletcher and W.F. Lindgren [Pacific J. Math. 43 (1971), 619–6311 on transitive quasi-uniformities and of S. Salbany [Thesis, Univ. Cape Town, 1971] on compactification and completion. The theorems as restated describe properties of certain right inverses of the functor which forgets the quasi-uniform structure and retains one induced topology (for Fletcher and Lindgren's work), respectively retains both induced topologies (for Salbany's work). Accordingly we investigate systematically the process by which the right inverses of the forgetful functors can be extended from the classical setting to one of these settings, and from one of these to the other.

On precompact (quasi-) uniform structures

The following are shown: (1) The subbasis theorem on precompactness does not hold even in uniform spaces. (2) The supremum of (even finitely many) precompact quasi-uniform structures is not necessarily precompact. (3) A compact quasi-uniform space is not necessarily totally bounded. These results contradict corresponding assertions in the literature.