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## Publications

Publications (66)

In [7] a relation between completeness of certain uniformity on ordered sets and restrictions of homeomorphisms of compactifications is described. We shall add more details here and correct one proof.

Conditions on equivalence of topological or uniform spaces having equivalent compactifications are investigated. For instance: (1) Let be Čech-complete, pseudoradial spaces complete in their compactifications . If the compactifications are homeomorphic then are homeomorphic. (2) Let be products of complete uniform spaces having linearly ordered bas...

The present paper generalizes to semitopological and quasitopological groups some results achieved by Horst Herrlich and the second author for topological groups. The results concern preserving products in coreflective subcategories. Unlike in paratopological or topological groups, there are non-finitely productive bicoreflective subcategories of q...

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...

Preserving products in some subcategories of paratopological groups is investigated. N. Noble proved in [6] that a product of sequential topological groups is sequential provided the cardinality of the index set of the product is a non-sequential cardinal. We shall generalize that result not only to paratopological groups but to any bicoreflective...

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].

We characterize Riesz spaces C(X) of real-valued continuous functions on a topological (Tychonoff) space X without using the Yosida representation theorem. The approach is elementary by using a simple description of zero-sets and a kind of local uniform completeness avoiding inversion closeness.

We obtain an internal characterization of the lattice U(X) of real-valued uniformly continuous functions on a uniform space X.

Our effort to weaken algebraic assumptions led us to obtain characterizations of C(X) as Riesz spaces, real ℓ-groups, semi-affine lattices and real lattices by using different techniques. We present a unified approach valid for any “convenient” category. By setting equivalent conditions to equi-uniform continuity, we obtain a characterization of th...

New couples of uniform spaces X, Y are found out for which a lattice isomorphism between U(X) and U(Y) implies a uniform homeomorphism between X and Y.

The set C(X)C(X) of real continuous functions on a topological space X has been characterized from several points of view depending on the algebraic structure considered on it. In this paper we propose a unified approach.

In our previous paper (Hušek and Pulgarín, Topol Appl, doi:10.1016/j.topol.2009.07.013, 2009) we characterized the set C(X) of real-valued continuous functions on a topological space X as a real ℓ-group. The present paper weakens the situation to the level of semi-affine lattices.
KeywordsSemi-affine lattice–Completely separating–Locally uniformly...

We provide a characterization of the lattice C(X) of real continuous functions on a completely regular space X. This result generalizes a solution of Anderson–Blair of the famous problem 81 of Birkhoff.

The lattice C(X) of all real-valued continuous functions on a topological space X has been characterized among various structures, for instance as an f-ring or as a Φ-algebra. In this paper, we characterize C(X) as a real ℓ-group (see Theorem 5.1).

The aim of this paper is to show a development of various methods of extensions of mappings and their interrelations. We shall point out some methods and relations entailing more general results than those originally stated.

Three approaches to a direct construction of Urysohn universal space are compared, namely those of Urysohn, Hausdorff and Katětov. More details are devoted to the unpublished Hausdorff's approach that is shown to work in a more general situation, too.

If sequential cardinals do not exist then every topological space is generated from a converging sequence by using finite
products, disjoint sums and quotients.

As applications of productivity of coreflective classes of topological spaces, the following results will be proved: (1) Characters of points of βN∖N are not smaller than any submeasurable cardinal less or equal to 2ω. (2) If κ is a submeasurable cardinal and S is a sequential fan with κ many spines then the tightness of the κ-power of S is equal t...

The κ-productivity of classes C of topological spaces closed under quotients and disjoint sums is characterized by means of Cantor spaces. The smallest infinite cardinals κ such that such classes are not κ-productive are submeasurable cardinals. It follows that if a class of topological spaces is closed under quotients, disjoint sums and countable...

Situations analogous to some classical characterization are investigated, of topological spaces X for which Cp(X) belongs to a given coreflective class C of locally convex spaces. For instance, if C contains all strong Mazur spaces and is contained in the class of weak Mazur spaces, then Cp(X) belongs to C iff X is realcompact. If C is the coreflec...

This paper is partly a survey on behavior of products in coreflective classes of some topological categories and partly it brings some new results completening those known. Classification: 54B10, 18B30, 03E10 We shall start with a motivation for our title and, after defining and explaining some basic notions, we concentrate on productivity of coref...

We prove that the countable product of supercomplete spaces having a countable closed cover consisting of partition-complete subspaces is supercomplete with respect to its metric-fine coreflection. Thus, countable products of σ-partition-complete paracompact spaces are again paracompact. On the other hand, we show (Theorem 7.5) that in arbitrary pr...

We investigate full subcategories of the category Ab of Abelian groups that are simultaneously reflective and coreflective in Ab. Such subcategories are exactly those isomorphic to categories of modules that are fully embedded into Ab. Rings giving rise to such modules are completely described. One of the curious special cases is provided by the fu...

A general version of the Stone–Weierstrass theorem is presented – one which involves no structure on the domain set of the real valued functions. This theorem is similar to the Stone–Weierstrass theorem which appears in the book by Gillman and Jerison, but instead of involving the concept of stationary sets the one presented here involves stationar...

It is shown that under a set-theoretical assumption, a product of Ulam nonmeasurably many topological linear spaces that are bornological with respect to all topological linear spaces, need not be bornological, thus contradicting a result of N. Adasch and A. P. Robertson.

H. Herrlich asked in Topology Appl.
49 (1993), 251–264, whether there are nontrivial classes of topological spaces that are almost reflective and almost coreflective at the same time. This question was dealt with (in Huek and Tozzi, Appl. Categ. Structures
4 (1996), 57–68) in a more general setting than almost reflective and almost coreflective cla...

Submeasurable cardinals are defined in a similar way as measurable cardinals are. Their characterizations are given by means of sequentially continuous pseudonorms (or homomorphisms) on topological groups and of sequentially continuous (or uniformly continuous) functions on Cantor spaces (for that purpose it is proved that if a complete Boolean alg...

For formulating results or definitions from older papers and books we shall mostly use modern terms and notation to avoid misunderstanding.

The Mackey theorem on products of bornological spaces is generalized to classes of topological linear spaces closed under quotients and inductive limits: either they are not countably productive or are nonmeasurably productive. The result can be shifted up to higher measurable cardinals. It is shown that the Mackey theorem is not valid for TLS-born...

It is shown that every continuous mapping from a metrizable space into a T0-space X can be factorized via a metrizable space of cardinality at most 2wX. The assumption of T0, is essential. This solves completely a Herrlich's problem about almost corefiectivity of metrizable spaces.

Sequentiality of products of topological groups is investigated.

An analogue of Kattov's theorem on the equality between the dimension of a Tychonov space and the analytic dimension of its ring of bounded real-valued continuous maps is established for proximity spaces and proximally continuous maps by an internal method of proof. A new kind of filter, called proximally prime filter, arises naturally as a tool in...

The Herrlich's problem from [8] whether there are nontrivial classes of topological spaces that are both almost reflective or injective and almost coreflective or projective, is investigated in a more general setting using cone and cocone modifications of the classes used in the problem. We look also at the problem for uniform spaces. Typical resul...

This paper summarizes the situation around the problem of when classes of projective objects are almost coreflective, both in general categories and in Top or similar categories. In addition to known results, several new contributions and examples are added.

In this paper, a pendant to a recent survey paper, the authors discuss several open problems in categorical topology. The emphasis is on topology-oriented problems rather than on more general category-oriented ones. In fact, most problems deal with full subconstructs or superconstructs of the constructTop of topological spaces and continuous maps.

Four levels of Galois connections are exhibited, starting with the classical one and going via concrete Galois connections to Galois adjunctions.

The structure of covers on subsets of products of metric spaces is investigated. Some applications to extensions of continuous maps and some well-known corollaries are given.

The notion of Galois connection will be used extensively in this book. Therefore, besides the definition, we report in this
chapter some basic results that will be used in later proofs. The reader who wishes to acquire further knowledge in this topic
could check [EKMS], for instance, where additional properties and many examples of Galois connectio...

Necessary and sufficient conditions for uniformities of uniform convergence on members of a collection to be realcompact are given. Theorems concerning commutation of products and the Hewitt realcompactification are proved by means of function spaces. As corollaries, some results concerning z-closed projections and products of k-spaces are obtained...

We provide an internal characterization of the sets C(X) of continuous real- valued functions on topological spaces X as real l-groups.

It is shown that productivity numbers of coreflective subcategories of topological linear spaces are precisely submeasurable cardinals (unlike locally con- vex spaces, where such numbers are measurable). A similar result is expected in topological spaces (only partial results are given here). Classification: 46A99, 54B10, 18B30, 03E10

"Papers from praque topsym 1991, held in praque, Czechoslovakia, Aug. 19-23, 1991" Incluye bibliografía e índice