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For a uniform space ( X , μ ) , we introduce a realcompactification of X by means of the family U μ ( X ) of all the real-valued uniformly continuous functions on X, in the same way that the known Samuel compactification of the space is given by U μ ⁎ ( X ) the set of all the bounded functions in U μ ( X ) . We will call it “the Samuel realcompactification” by several resemblances to the Samuel compactification. In this paper, we present different ways to construct such realcompactification as well as we study the corresponding problem of knowing when a uniform space is Samuel realcompact, that is, when it (topologically) coincides with its Samuel realcompactification. At this respect, we obtain as main result a theorem of Katětov–Shirota type, given in terms of a property of completeness, recently introduced by the authors, called Bourbaki-completeness.

Given a metric space X = ( X , d ) we show in ZF that: (a) The following are equivalent: (i) For every two closed and disjoint subsets A , B of X, d ( A , B ) > 0 . (ii) Every countable open cover of X has a Lebesgue number. (iii) Every real valued continuous function on X is uniformly continuous. (iv) For every countable (resp. finite, binary) open cover U of X, there exists a δ > 0 such that for all x , y ∈ X with d ( x , y ) δ , { x , y } ⊆ U for some U ∈ U . (b) If X is connected then: X is countably compact iff every open cover of X has a Lebesgue number iff for every two closed and disjoint subsets A , B of X, d ( A , B ) > 0 .

Let X = ( X , d ) , Y = ( Y , ρ ) be two metric spaces and f : X → Y be a function. f is called almost-bounded iff for every e > 0 there exists a δ > 0 such that for every A ⊆ X if the diameter of A is a ∈ A there exists b ∈ B with f ( a ) ∈ B ( b , e ) . We show that: (i) The image of a Cauchy sequence under a continuous and almost-bounded function has a Cauchy subsequence. (ii) The image of a totally bounded metric space under a continuous and almost-bounded function is totally bounded. (iii) If every continuous real valued function on a metric space is almost-bounded then the space is complete. The converse fails. (iv) A metric space is compact iff it is totally bounded and every continuous real valued function is almost-bounded. (v) Every continuous real valued function on a metric space is almost-bounded iff for every countable open cover of the space there exists a δ > 0 such that every subset of the space of diameter (vi) An open cover of a metric space has a Lebesgue number δ > 0 iff every two points of the space having distance

A space X is a Collins–Roscoe space if it has a countable point network satisfying the Collins–Roscoe structuring mechanism. In this article, we introduce a new property on the subspaces of product spaces called the minimal continuous factorization (MCF) property. We prove that if a subspace Y of the topological product of a family of cosmic spaces has the countable MCF property, then C p ( Y ) has the Collins–Roscoe property, hence it is meta-Lindelof. We also prove that, if X is a σ-compact metric space, then C p ( X ) satisfies σ-finite (F).

In the paper it is proved that there exists a continuous surjective mapping F : X → Y , where X and Y are separable complete metric spaces of transfinite dimension ind less than or equal to α d ∈ ω + ∪ { ∞ } and α r ∈ ω + ∪ { ∞ } , respectively, such that, for each continuous surjective mapping f : X → Y , where X and Y are compact metric spaces of transfinite dimension ind less than or equal to α d and α r , respectively, there are isometries i : X → X and j : Y → Y satisfying the relation F ∘ i = j ∘ f . This result remains true if we suppose that, for each mapping f : X → Y , Y coincides with a fixed separable complete metric space Y and that the corresponding mapping j : Y → Y is the identity.

We investigate the geometry of the family M of isometry classes of compact metric spaces, endowed with the Gromov–Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite metric spaces with the same number of points are isometric to some neighborhoods in the space R ∞ N , i.e., in the space R N with the norm ‖ ( x 1 , … , x N ) ‖ = max i ⁡ | x i | . As a corollary, we get that each finite metric space can be isometrically embedded into M in such a way that its image belongs to a subspace consisting of all finite metric spaces with the same number k of points. If the initial space has n points, then one can take k as the least possible integer with n ≤ k ( k − 1 ) / 2 .

Connectedness (resp. uniform connectedness [14] ) of uniform spaces can be defined in terms of continuous (resp. uniformly continuous) functions to a discrete space requiring that every continuous (resp. uniformly continuous) function to a discrete space has to be constant. Replacing uniformly continuous functions by the strictly weaker notion of Cauchy regular functions [15] we obtain a new notion of connectedness, namely Cauchy connectedness which happens to be an intermediate notion between connectedness and uniform connectedness. We primarily investigate several features of this new notion. Further in metric spaces, we turn our attention to quasi-Cauchy sequences [10] and show that replacing Cauchy regular continuity by ward continuity (functions preserving quasi-Cauchy sequences, see [11] ) one again gets back the notion of uniform connectedness. Finally we define WC spaces in line of UC spaces and obtain certain characterizations.

A metric space (X, d) is called finitely chainable if for every epsilon > 0, there are finitely many points p(1), p(2),..., p(r) in X and a positive integer m such that every point of X can be joined with some p(j), 1

For a metric space ( X , d ) , a subset A resolves ( X , d ) if each point x ∈ X is uniquely determined by the distances d ( x , a ) for a ∈ A . Also the metric dimension of ( X , d ) is the smallest cardinality m d ( X ) such that there is a set A of the cardinality m d ( X ) that resolves X. In this note we are going to determine the metric dimension of metric orbit spaces in some special cases and find an upper bound for a general case. This category contains a vast domain of topological spaces and topological manifolds.

We show that the negation of each one of the following statements is consistent with ZF: (i) Every sequentially compact metric space X = ( X , d ) is normal, i.e., the distance of any two disjoint non-empty closed subsets of X is strictly positive. (ii) If ( X , d ) is a sequentially compact metric space then X is a UC space, i.e., every continuous real valued on X is uniformly continuous. (iii) If ( X , d ) is a UC metric space then X is Lebesgue, i.e., every open cover of X has a Lebesgue number. (iv) If ( X , d ) is a metric space such that every countable open cover of X has a Lebesgue number then X is Lebesgue. We also show: (v) For every metric space X, the following are equivalent: (1) Every sequence in X admits a Cauchy subsequence; (2) For every sequence ( x n ) n ∈ N of X, for each e > 0 there is an infinite N e ⊆ N such that d ( x n , x m ) e whenever n , m ∈ N e ; (3) For every sequence ( x n ) n ∈ N of X, for each e > 0 and for each n 0 ∈ N there exist n , m ∈ N , n , m ≥ n 0 , n ≠ m such that d ( x n , x m ) e . (vi) The axiom of countable choice CAC implies that for every metric space X the following statements are equivalent: (1) X is Lebesgue; (2) Every countable open cover of X has a Lebesgue number; (3) X is UC.

For a Hausdorff Abelian topological group X, we denote by F 0 ( X ) the group of all X-valued null sequences endowed with the uniform topology. We prove that if X is an (E)-space (respectively, a strictly angelic space or a S-space), then so is F 0 ( X ) . We essentially simplify and clarify the theory of properties respected by the Bohr functor on Abelian topological groups, denoted below by X ↦ X + . We prove that for a complete maximally almost periodic group X, the group X shares with X + the same functionally bounded sets iff it shares the same compact sets and X + is a μ-space. We show that for a locally compact Abelian (LCA) group X the following are equivalent: 1) X is totally disconnected, 2) F 0 ( X ) is a Schwartz group, 3) F 0 ( X ) respects compactness, 4) F 0 ( X ) has the Schur property. So, if a LCA group X is not totally disconnected, the group F 0 ( X ) is a reflexive non-Schwartz group which does not have the Schur property. We prove also that for every compact connected metrizable Abelian group X the group F 0 ( X ) is monothetic and every real-valued uniformly continuous function on F 0 ( X ) is bounded.

We continue our investigation on those (partially) ordered metric spaces ( X , m , ≤ ) for which there exists a T 0 -quasi-metric d on X such that sup ⁡ { d , d − 1 } = m and for any x , y ∈ X we have that x ≤ y if and only if d ( x , y ) = 0 . In particular such partially ordered metric spaces are determined (in the sense of Nachbin) by a T 0 -quasi-uniformity with a countable base. We show that the compatibility conditions obtained in an earlier study between partial order and metric on the investigated spaces X become more transparent when additional compatibility conditions between metric and partial order with appropriate algebraic operations on X are assumed to hold.

Straight spaces are metric spaces X having the property that for a cover X = A ∪ B by two closed sets, any continuous function f : X → R is uniformly continuous provided it is so on each of the sets A and B (it is actually called ‘2-straight’ but is easier to deal with [8] ). In this paper we consider this nice idea and instead of uniformly continuous functions, we consider Cauchy regular functions [18] and ward continuous functions [12] , as these classes of functions strictly lie between the classes of continuous and uniformly continuous functions. In the process we obtain two natural variations of straightness which we name pre-straight and W-straight spaces respectively. We primarily investigate these notions along with another notion called pre ( ⁎ ) -straight which actually helps us to obtain a better understanding of the relation-ship between the notions of straight and pre-straight ness.

Let X = ( X , d , M , μ ) be a metric measure space, where d is a metric on X, M is a σ-algebra of X, and μ is a measure on M . Suppose that X satisfies the following conditions: (1) X is separable and locally compact; (2) M contains the Borel sets of X and for each E ∈ M , there exists a Borel set B ⊂ X such that E ⊂ B and μ ( B ∖ E ) = 0 ; (3) for every non-empty open set U ⊂ X , μ ( U ) > 0 ; (4) for all compact sets K ⊂ X , μ ( K ) ∞ ; and (5) X ∖ { x ∈ X | { x } ∈ M and μ ( { x } ) = 0 } is not dense in X. In this paper, we shall show that the space of real-valued uniformly continuous functions on X with the L p norm, 1 ≤ p ∞ , is homeomorphic to the subspace consisting of sequences conversing to 0 in the pseudo interior.

A topological group G is called M -factorizable if for every continuous real-valued function f : G → R on G, there exist a continuous homomorphism φ of G onto a first-countable topological group H and a continuous real-valued function g on H such that f = g ∘ φ . It is shown that an R -factorizable topological group is exactly an M -factorizable ω-narrow topological group. It is also proved that a subgroup H of an M -factorizable group G is M -factorizable if and only if H is z-embedded in G. We show that for a P-group G, G is M -factorizable if and only if every continuous real-valued function on G is uniformly continuous and G is ω-balanced. We also show that for a locally compact group G, G is M -factorizable if and only if G is either metrizable or σ-compact.

Let X, Y be complete metric spaces, U(X), U(Y) the spaces of uniformly continuous functions with real values defined on X and Y. We will show the form that every multiplicative isomorphism T:U(Y)→U(X) has: for x outside a uniformly isolated subset S⊂X, Tf(x)=sign(f(τ(x)))|f(τ(x))|1+p(x), where τ:X→Y is a uniform homeomorphism and p:X∖S→R is such that p⋅h⋅log⁡h is uniformly continuous for every h∈U(X,(2,∞)).

Several Banach–Stone-like generalizations of Shirota's result for metrizable uniform spaces are proved. Namely, if complete uniform spaces X , Y have isomorphic lattices U ( X ) , U ( Y ) of their real-valued uniformly continuous functions, and both X , Y are either some products of spaces having monotone bases (metrizable or uniformly zero-dimensional), or are locally fine and of non-measurable cardinality, then X and Y are uniformly homeomorphic.

If two uniform spaces have isomorphic lattices of their uniformly continuous real-valued functions then also their sublattices of bounded functions are isomorphic. That result is used to give a different correct proof of Shirota theorem (complete metric spaces are determined by their uniformly continuous real-valued functions) than that in [1] .

In this paper, we consider spaces having the so-called property of f -distances, where f is a positive decreasing function defined on ω such that f ( n ) ≤ 1 2 n . It is proved that for well-known classes S of separable metric spaces (in [2] they are called isometrically ω -saturated classes of spaces) the following is true: for a given collection S of elements of S with the property of f -distances, there exists an element of S with the property of g -distances containing isometrically each element of S , where g is the function on ω for which g ( n ) = f ( n + 2 ) , n ∈ ω .

A simplified variant of the definition of the completeness for metric mappings (that is closed to the standard definition of the completeness for metric spaces) is obtained. This result is used to construct the completions of metric mappings by a method close to the standard completion method for metric spaces. Relations between completions, fibrewise completions and fibrewise complete extensions of metric mappings are clarified. It is shown that for closed metric mappings all of these extensions coincide (since the completeness of these mappings coincides with their fibrewise completeness). The Lavrentieff theorem (about G δ -extensions of homeomorphisms between subsets of metric spaces) is extended to metric mappings. It is proved that a uniformly continuous map-morphism of metric mappings may be extended to a uniformly continuous map-morphism of their completions. The end of the paper contains generalizations of the Nagata–Smirnov and Bing metrization theorems for mappings.

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is compact-preserving if and only if for each point $x\in X$ there is a compact subset $K_x\subset Y$ such that for each neighborhood $O_{f(x)}\subset Y$ of $f(x)$ there is a neighborhood $O_x\subset X$ of $x$ such that $f(O_x)\subset O_{f(x)}\cup K_x$ and the set $K_x\setminus O_{f(x)}$ is finite. This characterization is applied to give an alternative proof of a classical characterization of continuous functions on locally connected metrizable spaces as functions that preserve compact and connected sets. Also we show that for each compact-preserving function $f:X\to Y$ defined on a (strong) Fr\'echet space $X$, the restriction $f|LI'_f$ (resp. $f|LI_f)$ is continuous. Here $LI_f$ is the set of points $x\in X$ of local infinity of $f$ and $LI'_f$ is the set of non-isolated points of the set $LI_f$. Suitable examples show that the obtained results cannot be improved., Comment: 6 pages

In this paper, we show that fixed point generalizations to partial metric spaces can be obtained from the corresponding results in metric spaces. We consider some cases to demonstrate this fact.

We show in ZF that: (i) A countably compact metric space need not be limit point compact or totally bounded and, a limit point compact metric space need not be totally bounded. (ii) A complete, totally bounded metric space need not be limit point compact or Cantor complete. (iii) A Cantor complete, totally bounded metric space need not be limit point compact. (iv) A second countable, limit point compact metric space need not be totally bounded or Cantor complete. (v) A sequentially compact, selective metric space (the family of all non-empty open subsets of the space has a choice function) is compact. (vi) A countable product of sequentially compact (resp. compete and totally bounded) metric spaces is sequentially compact (resp. compete and totally bounded).

According to Katětov (1988) [15] , for every infinite cardinal m satisfying m n ⩽ m for all n m , there exists a unique m -homogeneous universal metric space U m of weight m . This object generalizes the classical Urysohn universal metric space U = U ℵ 0 . We show that for m uncountable, the isometry group Iso ( U m ) with the topology of simple convergence is not a universal group of weight m : for instance, it does not contain Iso ( U ) as a topological subgroup. More generally, every topological subgroup of Iso ( U m ) having density m and possessing the bounded orbit property ( OB ) is functionally balanced: right uniformly continuous bounded functions are left uniformly continuous. This stands in sharp contrast with Uspenskijʼs (1990) [35] result about the group Iso ( U ) being a universal Polish group.

We give a survey of Jun-iti Nagataʼs many contributions to dimension theory: characterizations of n -dimensional metric spaces in terms of a special base; characterizations of n -dimensional metrizable spaces in terms of a special metric; imbedding theorems and universal spaces for n -dimensional metric spaces; countable-dimensional metric spaces; dimension and rings of continuous functions; dimension theory beyond metric spaces.

The general condition that characterizes the possibility of actionʼs extension over a completion of a phase space is formulated and an example of a G -space that has no Dieudonne complete G -extensions is given. Sufficient conditions (different kinds of rectangular conditions in products) for the actionʼs extensions over the Stone–Cech compactification, the Hewitt realcompactification and the Dieudonne completion of a space are presented. Boundedness, uniform equicontinuity and quasiboundedness of actions are characterized as actionʼs uniform continuity on the (piecewise) semi-uniform product. From this point of view the origin of different examples of actionʼs extensions are explained.

Let (X,τ) be a topological space and let ρ be a metric defined on X. We shall say that (X,τ) is fragmented by ρ if whenever ε>0 and A is a nonempty subset of X there is a τ-open set U such that U∩A≠∅ and ρ−diam(U∩A)

We look at the dynamics of continuous self-maps of compact metric spaces possessing the pseudo-orbit tracing property (i.e., the shadowing property). Among other things we prove the following: (i) the set of minimal points is dense in the non-wandering set Ω ( f ) , (ii) if f has either a non-minimal recurrent point or a sensitive minimal subsystem, then f has positive topological entropy, (iii) if X is infinite and f is transitive, then f is either an odometer or a syndetically sensitive non-minimal map with positive topological entropy, (iv) if f has zero topological entropy, then Ω ( f ) is totally disconnected and f restricted to Ω ( f ) is an equicontinuous homeomorphism.

We prove that there is a residual subset of the Gromov–Hausdorff space ( i.e. the space of all compact metric spaces up to isometry endowed with the Gromov–Hausdorff distance) whose elements enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.

We consider a geodesically complete and proper Hadamard metric measure space X endowed with a Borel measure. Assuming that there exists a certain non-amenable group of isometry of X which acts freely, properly discontinuously and cocompactly on X and preserves the measure we show that the topological entropy of the geodesic flow on the orbit space is positive.

International audience; Let $K$ be an ultrametric complete field and let $E$ be an ultrametric space. Let $A$ be the Banach $K$-algebra of bounded continuous functions from $E$ to $K$ and let $B$ be the Banach $K$-algebra of bounded uniformly continuous functions from $E$ to $K$. Maximal ideals and continuous multiplicative semi-norms on $A$ (resp. on $B$) are studied by defining relations of stickness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of $A$ (resp. of $B$) is in bijection with the set of equivalence classes with respect to stickness (resp. to contiguousness). Every prime ideal of $A$ or $B$ is included in a unique maximal ideal and every prime closed ideal of $A$ (resp. of $B$) is a maximal ideal, hence every continuous multiplicative semi-norms on $A$ (resp. on $B$) has a kernel that is a maximal ideal. If $K$ is locally compact, every maximal ideal of $A$, (resp. of $B$) is of codimension $1$. Every maximal ideal of $A$ or $B$ is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on $E$. Consequently, on $A$ as on $B$ the set of continuous multiplicative semi-norms defined by points of $E$ is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of $A , \ Max(A)$ and the Banaschewski compactification of $E$ which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of $A$ (resp. $B$) is equal to the whole set of continuous multiplicative semi-norms.

Let M be a uniform space and X the Banach space of bounded and uniformly continuous functions from M into R , provided with its supremum norm. The aim of this paper is to discuss the connection between the geometry of X and the nature of M. In particular, we will prove that certain reconstructions of the unit ball of X by means of its extreme points admit a translation in terms of extension of uniformly continuous functions. We also analyze the impact of these properties on the Samuel compactification of M.

The following result due to Hanai, Morita, and Stone is well known: Let f be a closed continuous map of a metric space X onto a topological space Y. Then the following statements are equivalent: (i) Y satisfies the first countability axiom; (ii) for each y ∈ Y , f − 1 { y } has a compact boundary in X; (iii) Y is metrizable. In this article we obtain several related results in the setting of topological ordered spaces. In particular we investigate the upper and lower topologies of metrizable topological ordered spaces which are both C- and I-spaces in the sense of Priestley.

Given a compact metric space X and a continuous map f from X to itself, we construct a barrier function for chain-recurrence. We use it to endow the space of chain-transitive components with a non-trivial ultrametric distance and to construct Lyapunov functions for f . Most of these constructions are then generalized on an arbitrary separable metric space to a continuous compactum-valued map.

The main aim of this paper is to give a positive answer to a question of Behrends, Geschke and Natkaniec regarding the existence of a connected metric space and a non-constant real-valued continuous function on it for which every point is a local extremum. Moreover we show that real-valued continuous functions on connected spaces such that every family of pairwise disjoint non-empty open sets is of size | R | are constant provided that every point is a local extremum.

A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph R, the universal Urysohn metric space U, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of U (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides extend to affine representations of various isometry groups. As an application of this technique, we show that U admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.

The paper establishes, within constructive mathematics, a full and faithful functor M from the category of locally compact metric spaces and continuous functions into the category of formal topologies (or equivalently locales). The functor preserves finite products, and moreover satisfies f⩽g if, and only if, M(f)⩽M(g) for continuous f,g:X→R. This makes it possible to transfer results between Bishop's constructive theory of metric spaces and constructive locale theory.

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. The uniform universal cover of a coverable space is a kind of generalized cover with universal and lifting properties in the category of uniform spaces and uniformly continuous mappings. Associated with the uniform universal cover is a functorial uniform space invariant called the deck group , which is related to the classical fundamental group by a natural homomorphism. We obtain some specific results for one-dimensional spaces.

In this paper we prove a theorem more general than the following. Suppose that X is Cech-complete and Y is a closed subset of a product of a separable metric space with a compact Hausdorff space. Then for each separately continuous function f : X × Y → R there exists a residual set R in X such that f is jointly continuous at each point of R × Y . This confirms the suspicions of S. Mercourakis and S. Negrepontis from 1991.

This paper develops the basic theory of quotients of uniform spaces via sufficiently nice group actions. We generalize and unify two fundamental constructions: quotients of topological groups via closed normal subgroups and quotients of metric spaces via actions by isometries. Basic results about inverse limits of topological groups are extended to inverse limits of group actions on uniform spaces, and notions of prodiscrete action and generalized covering map are introduced.

Let G be a Hausdorff topological group. It is shown that there is a class C of subspaces of G, containing all (but not only) precompact subsets of G, for which the following result holds: Suppose that for every real-valued discontinuous function on G there is a set A ∈ C such that the restriction mapping f | A has no continuous extension to G; then the following are equivalent: (i) the left and right uniform structures of G are equivalent, (ii) every left uniformly continuous bounded real-valued function on G is right uniformly continuous, (iii) for every countable set A ⊂ G and every neighborhood V of the unit e of G, there is a neighborhood U of e in G such that A U ⊂ V A . As a consequence, it is proved that items (i), (ii) and (iii) are equivalent for every inframetrizable group. These results generalize earlier ones established by Itzkowitz, Rothman, Strassberg and Wu, by Milnes and by Pestov for locally compact groups, by Protasov for almost metrizable groups, and by Troallic for groups that are quasi-k-spaces.

For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent.

We consider metric spaces X with the nice property that any continuous function f :X→ R which is uniformly continuous on each set of a finite cover of X by closed sets, is itself uniformly continuous. We characterize the spaces with this property within the ample class of all locally connected metric spaces. It turns out that they coincide with the uniformly locally connected spaces, so they include, for instance, all topological vector spaces. On the other hand, in the class of all totally disconnected spaces, these spaces coincide with the UC spaces.

A set H of continuous mappings from a topological space X to a uniform space (Y,U) is said to be functionally equicontinuous (at x∈X) if the set {f∘h:h∈H} is equicontinuous (at x) for each bounded real-valued uniformly continuous function f on (Y,U). In this paper, information about this concept is given. The relation between equicontinuity (at x∈X) and functional equicontinuity (at x∈X) is examined in detail. The main result asserts that for every X belonging to a wide class C of topological spaces (including all quasi-kR-spaces), any set of continuous mappings from X to any uniform space Y which is functionally equicontinuous is in fact equicontinuous. Applications to topological groups of the general results are given in the last section. In particular, the main result is applied to solve positively in the class C the problem of the equality [FSIN]=[SIN] raised by Itzkowitz.

Let X be a set and F a family of real-valued functions on X. We denote by μFX the space X endowed with the weak uniformity given by F. In this paper we provide a method of generating the set U(μFX), of all uniformly continuous real functions on μFX, by means of the family F. In order to do that we need to study the uniform approximation of real uniformly continuous functions on subsets of Rn. As a consequence, we give an internal condition on F in order to be uniformly dense in U(μFX).

Given classes of domains (or topological spaces) A and B, when are all function spaces [A→B] again continuous domains? The principle result of this paper is that for A either all compact and core compact spaces or only the single domain consisting of a decreasing sequence with two lower bounds, then the largest B consists of all continuous domains such that ↓x is a sup-semilattice for each x. We also establish an analogue for L-domains.

We prove that every complete metric space X that is thin (i.e., every closed subspace has connected uniform quasi components) has the compact separation property (for any two disjoint closed connected subspaces A and B of X there is a compact set K disjoint from A and B such that every neighbourhood of K disjoint from A and B separates A and B ). The real line and all compact spaces are obviously thin. We show that a space is thin if and only if it does not contain a certain forbidden configuration. Finally we prove that every metric UA -space (see [Rend. Instit. Mat. Univ. Trieste 25 (1993) 23–56]) is thin. The UA -spaces form a class properly including the Atsuji spaces.

We characterize a metric uc-ness of local nature, uc-ness means some continuity is uniform, as a uniform separation property. Then we reformulate it as a relationship between hypertopologies and finally as agreement between function space topologies.

As a generalization of developments of (developable) spaces, we introduce the notion of σ -strong networks of spaces, and related notions. We give new characterizations for around quotient compact images of metric spaces by means of σ -strong networks, and give some related matters.

We study properties of Hausdorff spaces X which depend on the variety of continuous selections for their Vietoris hyperspaces F (X) of closed non-empty subsets. Involving extreme selections for F (X) , we characterize several classes of connected-like spaces. In the same way, we also characterize several classes of disconnected-like spaces, for instance all countable scattered metrizable spaces. Further, involving another type of selections for F (X) , we study local properties of X related to orderability. In particular, we characterize some classes of orderable spaces with only one non-isolated point.