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The (dis)connectedness of products of Hausdorff spaces in the box topology

Vitalij A. Chatyrko (2021)

Commentationes Mathematicae Universitatis Carolinae

In this paper the following two propositions are proved: (a) If X α , α ∈ A , is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product □ α ∈ A X α can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If X α , α ∈ A , is an infinite system of Brown Hausdorff topological spaces then the box product □ α ∈ A X α is also Brown Hausdorff, and hence, it is connected. A space is Brown if...

The disjoint arcs property for homogeneous curves

Paweł Krupski (1995)

Fundamenta Mathematicae

The local structure of homogeneous continua (curves) is studied. Components of open subsets of each homogeneous curve which is not a solenoid have the disjoint arcs property. If the curve is aposyndetic, then the components are nonplanar. A new characterization of solenoids is formulated: a continuum is a solenoid if and only if it is homogeneous, contains no terminal nontrivial subcontinua and small subcontinua are not ∞-ods.

The Doitchinov Completion of a Regular Paratopological Group

Künzi, Hans-Peter, Romaguera, Salvador, Sipacheva, Ol’ga (1998)

Serdica Mathematical Journal

In memory of Professor D. Doitchinov ∗ This paper was written while the first author was supported by the Swiss National Science Foundation under grants 21–30585.91 and 2000-041745.94/1 and by the Spanish Ministry of Education and Sciences under DGES grant SAB94-0120. The second author was supported under DGES grant PB95-0737. During her stay at the University of Berne the third author was supported by the first author’s grant 2000-041745.94/1 from the Swiss National Science Foundation.We show...

The dual form of Knaster-Kuratowski-Mazurkiewicz principle in hyperconvex metric spaces and some applications

George Isac, George Xian-Zhi Yuan (1999)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, we first establish the dual form of Knaster- Kuratowski-Mazurkiewicz principle which is a hyperconvex version of corresponding result due to Shih. Then Ky Fan type matching theorems for finitely closed and open covers are given. As applications, we establish some intersection theorems which are hyperconvex versions of corresponding results due to Alexandroff and Pasynkoff, Fan, Klee, Horvath and Lassonde. Then Ky Fan type best approximation theorem and Schauder-Tychonoff fixed point...

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

Throughout this abstract, G is a topological Abelian group and G ^ is the space of continuous homomorphisms from G into the circle group 𝕋 in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism G ^ ↠ D ^ given by h ↦ h | D is a homeomorphism, and G is determined if each dense subgroup of G determines G . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...

The dual space of precompact groups

M. Ferrer, S. Hernández, V. Uspenskij (2013)

Commentationes Mathematicae Universitatis Carolinae

For any topological group G the dual object G ^ is defined as the set of equivalence classes of irreducible unitary representations of G equipped with the Fell topology. If G is compact, G ^ is discrete. In an earlier paper we proved that G ^ is discrete for every metrizable precompact group, i.e. a dense subgroup of a compact metrizable group. We generalize this result to the case when G is an almost metrizable precompact group.

The Dugundji extension property can fail in ωµ -metrizable spaces

Ian Stares, Jerry Vaughan (1996)

Fundamenta Mathematicae

We show that there exist ω μ -metrizable spaces which do not have the Dugundji extension property ( 2 ω 1 with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.

The elementary-equivalence classes of clopen algebras of P-spaces

Brian Wynne (2008)

Fundamenta Mathematicae

Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.

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