Quasi-uniform Space

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.

The present paper is concerned with the Ekeland Variational Principle (EkVP) and its equivalents (Caristi–Kirk fixed point theorem, Takahashi minimization principle, Oettli-Théra equilibrium version of EkVP) in quasi-uniform spaces. These extend some results proved by Hamel and Löhne [A minimal point theorem in uniform spaces. In: Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vols 1, 2. Dordrecht: Kluwer Academic Publishers; 2003. p. 577–593] and Hamel [Equivalents to Ekeland's variational principle in uniform spaces. Nonlinear Anal. 2005;62:913–924] in uniform spaces, as well as those proved in quasi-metric spaces by various authors. The case of F-quasi-gauge spaces, a non-symmetric version of F-gauge spaces introduced by Fang [The variational principle and fixed point theorems in certain topological spaces. J Math Anal Appl. 1996;202:398–412], is also considered. The paper ends with the quasi-uniform versions of some minimization principles proved by Arutyunov and Gel'man [The minimum of a functional in a metric space, and fixed points. Zh Vychisl Mat Mat Fiz. 2009;49:1167–1174] and Arutyunov [Caristi's condition and existence of a minimum of a lower bounded function in a metric space. Applications to the theory of coincidence points. Proc Steklov Inst Math. 2015;291(1):24–37] in complete metric spaces.

e-9 - Quasi-Uniform Spaces

In this paper, introducing generalized quasi uniformity (in short, \(g\)-quasi uniformity) and \(g\)-quasi uniformly continuous maps, it has been shown that every supratopology is achievable from a \(g\)-quasi uniformity; moreover, a \(g\)-continuous map between a pair of supratopological spaces is indeed the induced map obtained from the \(g\)-quasi uniformly continuous map between the corresponding \(g\)-quasi uniform spaces. These results establish a functorial correspondence between the respective categories. Descriptions of \(g\)-neighbourhood system, \(g\)-interior operator and subspaces of GTS alongwith a characterization of \(\mu\)-\(T_1\)-ness of a GTS in terms of \(g\)-quasi uniformity are established. Further, introducing \(g\)-quasi uniform isomorphism in a natural way, a couple of \(g\)-quasi uniform properties are also discussed.

Completion of continuity spaces with uniformly vanishing asymmetry

Functorial admissible quasi-uniformities on topological spaces

Incomplete infima of complete uniformities related to topological groups and topological vector spaces

It is shown that if ( X , δ ) (X,\delta ) is an Efremovič proximity space, Y is a topological space, R ⊂ X × Y R \subset X \times Y is an injective relation, then R is open if and only if R is weakly open, nearly open and R [ X ] R[X] is open in Y. An analogous result is proved when X is a uniform space and Y a Morita uniform space: (i) if R is uniformly open, then R is weakly open and uniformly nearly open; (ii) if R is weakly open, and uniformly nearly open, then R is uniformly open on X to R [ X ] R[X] . These results include, as special cases, results of Kelley, Pettis and Weston.

The cartesian closed topological (CCT) hull of a concrete categoryK is the smallest full CCT extension ofK. A general method for describing the CCT hulls is presented and applied to render unified proofs for the basic examples ofK: topological spaces, uniform spaces, pretopological spaces, compact T2 spaces, metrizable spaces, and completely regular spaces.

In strictly two-sided, commutative biquantale, we show that Hutton (L, <TEX>$\otimes$</TEX>)-uniform spaces and (L, <TEX>$\odot$</TEX>)-uniform spaces induce enriched (L, <TEX>$\odot$</TEX>)-topological spaces and enriched (L, <TEX>$\odot$</TEX>)-interior spaces.

First, we discuss the behavior of boundedness in uniform spaces with respect to subspaces, projective limits, and suprema in relation to precompactness. A special uniformly isomorphic embedding of un arbitrary uniform space in a bounded uniform space is presented and examined in 2.6. Hejcman’s characterization (by B-conservativity) of uniform spaces in which boundedness can be tested by a single pseudometric is proved in a new way, see 3.13, using a version 3.1 of the metrization lemma. We comment briefly on boundedness in topological vector spaces. In topological groups we investigate a hierarchy of partly new notions of boundedness, strongly interrelated among themselves, and exhibit various situations in which certain of these notions coincide. Boundedness respecting subspaces of a uniform space prove useful. Many examples illustrate and complement the general theory, see, e.g., Example 6.4.

In this note we prove the following.

The usual ‘closed-graph theorems’ are concerned with the question of which topological structures on sets X, Y will ensure equivalence of continuity and closed-graph conditions for certain mappings t: X → Y. In this paper some stronger closed-graph like conditions are introduced on a mapping between uniform spaces, by considering the ‘hypergraph’, or graph of the induced mapping between hyperspaces. The central result, included in Theorem 1, states that for arbitrary uniform spaces X, Y, t is continuous if its hypergraph is closed. Thus for topological vector spaces (and some others) the closed-hypergraph condition is equivalent to continuity. In section 2 some situations are found in which continuity is implied by certain intermediate conditions to closed-graph and closed-hypergraph conditions. In particular, a closed-hypergraph theorem is shown to hold for locally convex spaces, provided only that X is barrelled. Finally, in section 3, the hypergraph of a relation is studied, and the separatedness of the quotient space of an equivalence relation is shown to replace continuity with regard to closed-graph and closed-hypergraph conditions.

Uniform spaces are topological spaces with additional structure. In this paper, we describe the uniform structure of a uniform space as a multi-context and we discuss some properties of a uniform space in the language of Formal Concept Analysis. In the second part of the paper, we define a uniform structure between two sets, G and M, called contextual uniformity and we discuss some topological aspects related to this construction.KeywordsUniform spacemulti-contexttopologyFormal Concept Analysis

All topological spaces shall be uniformizable (completely regular Hausdorff). A uniformity on X shall be viewed as a collection μ of coverings of X, via the manner of Tukey [20] and Isbell [16], and the associated uniform space denoted μX. Given the uniformizable topological space X, we shall be concerned with compatible uniformities as follows (discussed more carefully in § 1). The fine uniformity α (finest compatible with the topology); the “cardinal reflections“ αm of α (m an infinite cardinal number) ; αc, the weak uniformity generated by the real-valued continuous functions.With μ standing, generically, for one of these uniformities, we consider the question: when is μ(X × Y) = μX × μY For μ = αℵ0 (the finest compatible precompact uniformity), the problem is equivalent to that of whenβ(X × Y) = βX × βY,β denoting Stone-Cech compactification; this is answered by the theorem of Glicksberg [9]. For μ = α, we have Isbell's generalization [16, VI1.32].