Completion of quasi-topological groups

A topology of a quasi-topological group is induced by several natural semi-uniformities, namely right, left, two-sided and Roelcke semi-uniformities. A quasi-topological group is called complete if every Cauchy (in some sense—we examine several generalizations of Cauchy properties) filter on the two-sided semi-uniformity converges.We use the theory of Hausdorff complete semi-uniform spaces, see [B. Batíková, Completion of semi-uniform spaces, Appl. Categor. Struct. 15 (2007) 483-491], and show that Hausdorff complete quasi-topological groups form an epireflective subcategory of Hausdorff quasi-topological groups. But the reflection arrows need not be embeddings.For several types of Cauchy-like properties we show examples of quasi-topological groups that cannot be embedded into a complete group.     

Completion of merotopic spaces and extension of uniformly continuous maps

A general Riesz merotopic space (X, ν) determines a not necessarily topological closure operator c_ν on X. The space (X, ν) is said to be complete if every cluster on (X, ν) is contained in an adherence grill on (X, c_ν). We discuss a method of obtaining a large class of completions of a given Riesz merotopic space with induced T₁ closure space. As special cases we get the simple completion, which induces a simple closure space extension, and the strict completion, which induces a strict closure space extension. We show that the category of complete separated T₁ Riesz merotopic spaces is epireflective in the category of separated T₁ Riesz merotopic spaces, the reflection of an object being the simple completion. Similarly the category of complete clan-covered quasi-regular T₁ Riesz merotopic spaces is epireflective in the category of clan-covered quasi-regular T₁ Riesz merotopic spaces, the reflection of an object being the strict completion.     

Hausdorff continuous interval-valued functions and quasicontinuous functions

In the paper it is shown that every Hausdorff continuous interval-valued function corresponds uniquely to an equivalence class of quasicontinuous functions. This one-to-one correspondence is used to construct the Dedekind order completion of C(X), the set of all real-valued continuous functions, when X is a compact Hausdorff topological space or a complete metric space.     

An elementary approach to ‘Algebra ∩ topology = compactness’

Herrlich and Strecker characterized the category Comp ₂ of compact Hausdorff spaces as the only nontrivial full epireflective subcategory in the category Top ₂ of all Hausdorff spaces that is concretely isomorphic to a variety in the sense of universal algebra including infinitary operations. The original proof of this result requires Noble's theorem, i.e. a space is compact Hausdorff iff every of its powers is normal, which is far from being elementary. Likewise, Petz' characterization of the class of compact Hausdorff spaces as the only nontrivial epireflective subcategory of Top ₂, which is closed under dense extensions (= epimorphisms in Top ₂) and strictly contained in Top ₂ is based on a result by Kattov stating that a space is compact Hausdorff iff its every closed subspace is H-closed. This note offers an elementary approach for both, instead.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

On completion of fuzzy metric spaces

Completions of fuzzy metric spaces (in the sense of George and Veeramani) are discussed. A complete fuzzy metric space Y is said to be a˜fuzzy metric completion of a˜given fuzzy metric space X if X is isometric to a˜dense subspace of Y. We present an example of a˜fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a˜dense subspace uniformly isomorphic to it.     

Closed structures on categories of topological spaces

The category of all topological spaces and continuous maps and its full subcategory of all T_o-spaces admit (up to isomorphism) precisely one structure of symmetric monoidal closed category (see [2]). In this paper we shall prove the same result for any epireflective subcategory of the category of topological spaces (particularly e.g. for the categories of Hausdorff spaces, regular spaces, Tychonoff spaces).     

Essential extensions of Hausdorff spaces

In the category Haus of Hausdorff spaces the only injectives are the one-point spaces. Even though every Hausdorff spaceX has a maximal essential extension,X fails to have an injective hull, providedX has more than one point. A non-empty Hausdorff space has a proper essential extension if and only ifX is locally H-closed but not H-closed. In this case,X has (up to isomorphism) precisely one proper essential extension: the Obreanu-Porter extension (being simultaneously its maximal essential extension and its minimal H-closed extension). Completely parallel results hold for the categories SReg, Reg, and Tych of semi-regular, regular, and completely regular spaces respectively. In particular, the Alexandroff compactifications of locally compact, non-compact Hausdorff spaces are characterized categorically as the proper essential extensions of non-empty spaces in Tych (resp. Reg).Dedicated to my friend Nico Pumplün on his sixtieth birthday     

Asymmetry and bicompletion of approach spaces

We develop a bicompletion theory for the category Ap₀ of T₀ approach spaces in the sense of Lowen [R. Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford University Press, Oxford, 1997], which extends the completion theory obtained in [R. Lowen, K. Robeys., Completions of products of metric spaces, Quart. J. Math. Oxford 43 (1991) 319-338] for the subcategory of Hausdorff uniform approach spaces. Moreover, we prove it to be firmly epireflective (in the sense of [G.C.L. Brümmer, E. Giuli, A categorical concept of completion of objects, Comment. Math. Univ. Carolin. 33 (1992) 131-147]) with respect to a certain morphism class of dense embeddings.     

Epireflective subcategories of Hausdorff categories

Let pHaus denote the category of Hausdorff spaces and p-maps, and let HCL denote the subcategory of pHaus consisting of H-closed spaces and continuous functions. It is well-known that HCL is an epireflective subcategory of pHaus. In this paper we characterize the epireflective subcategories of pHaus that contain HCL.     

Merotopological Spaces

In this paper we introduce a new topological-type of structured set called merotopological space. The appropriate morphisms are defined and characterizations of the corresponding initial and final structures are given. The resulting category contains as fully embedded subcategories not only the category of topological spaces and continuous maps but also the category of merotopic spaces and uniformly continuous maps, and, a fortiori, the category of nearness spaces and the category of uniform spaces. A functorial completion is constructed for merotopological spaces using bunches. A problem that has remained long open in the setting of nearness spaces is to find an internal characterization of the epireflective hull of the topological spaces. We solve the analogue of this problem in the setting of merotopological spaces. Applications to the Wyler prime closed filter compactification and to Taimanov's extension theorem are given.     

Approximations from Subspaces of C0(X)

We show among other things that if B is a linear space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff space X, for which there is a continuous function h defined in a neighbourhood of 0 in the real line which is non-affine in every neighbourhood of 0 and satisfies |h(t)|k |t| for all t, such that hb is in B whenever b is in B and the composite function is defined, then every function in C₀(X) which can be approximated on every pair of points in X by functions in B can be approximated uniformly by functions in B.     

Local connectedness and extension of uniformly continuous functions

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. The straight spaces have been studied in [A. Berarducci, D. Dikranjan, J. Pelant, An additivity theorem for uniformly continuous functions, Topology and its Applications 146-147 (2005) 339-352], which contains characterization of the straight spaces within the class of the locally connected spaces (they are the uniformly locally connected ones) and the class of the totally disconnected spaces (they coincide with the totally disconnected Atsuji spaces). We show that the completion of a straight space is straight and we characterize the dense straight subspaces of a straight space. In order to clarify further the relation between straightness and the level of local connectedness of the space we introduce two more intermediate properties between straightness and uniform local connectedness and we give various examples to distinguish them. One of these properties coincides with straightness for complete spaces and provides in this way a useful characterization of complete straight spaces in terms of the behaviour of the quasi-components of the space.     

Hilbert-generated spaces

We classify several classes of the subspaces of Banach spaces X for which there is a bounded linear operator from a Hilbert space onto a dense subset in X. Dually, we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in Hilbert spaces or in c₀(Γ) spaces in their weak topology. The existence of such embeddings is characterized by the existence of certain uniformly Gâteaux smooth norms.     

Products of Compacta with Plyhedra and Topological Spaces in the Shape Category

In the literature there exist examples of separable metric spaces X,Y whose Cartesian product X × Y is not a product in the shape category Sh(Top). It is an open question whether, for X a compact Hausdorff space, X × Y is a product in Sh(Top), for every topological spaces Y. The main result of the paper asserts that the answer is positive provided X × P is a product in Sh(Top), for every polyhedron P.     

Generic existence of a nonempty compact set of fixed points

Let X be a complete metric space, a set of continuous mappings from X into itself, endowed with a metric topology finer than the compact-open topology. Assuming that there exists a dense subset contained in such that for every mapping T in the set {x ε X: Tx = x} is nonempty, it is proved that most mappings (in the Baire category sense) in do have a nonempty compact set of fixed points. Some applications to α-nonexpansive operators, semiaccretive operators and differential equations in Banach spaces are derived.     

Local Uniform Strong Proximinality of the Space of Continuous Vector-Valued Functions

Let X be a Banach space, S be a compact Hausdorff space and Y be a U-proximinal subspace of X. We prove that C(S,Y) is locally uniformly strongly proximinal in C(S,X) and the corresponding metric projection map is Hausdorff metric continuous.     

A Convenient Subcategory of Tych

A map f:XY between Hausdorff topological spaces is k-continuous if its restriction f| _K to every compact subspace K of X is continuous. X is called a k _R -space if every k-continuous function from X to a Tychonoff space is continuous. In this paper we investigate the category of Tychonoff k _R -spaces, and show that it is Cartesian closed (thus convenient in the sense of Wyler).     

An Isometric Representation of the Dual of {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)}

Many structures in functional analysis are introduced as the limit of an inverse (aka projective) system of seminormed spaces [2, 3, 8]. In these situations, the dual is moreover equipped with a seminorm. Although the topology of the inverse limit is seldom metrizable, there is always a natural overlying locally convex approach structure. We provide a method for computing the adjoint of this space, by showing that the dual of a limit of locally convex approach spaces becomes a co-limit in the category of seminormed spaces. As an application we obtain an isometric representation of the dual space of real valued continuous functions on a locally compact Hausdorff space X, equipped with the compact open structure.     

The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X,K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X,K) is a bifunctor. Using this result, it is proved that the Cartesian product X×Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt)×Sh(Top)→Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X×Z need not be a direct product in Sh(Top).