Continuous Selections and Locally Pseudocompact Groups (original) (raw)
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On zero-dimensionality and the connected component of locally pseudocompact groups
Proceedings of the American Mathematical Society
A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is dense in the component of the completion of G, and G/G_0 is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that G/G_0 may fail to be zero-dimensional even for totally minimal pseudocompact groups.
CLP-compactness for topological spaces and groups
Topology and its Applications, 2007
We study CLP-compact spaces (every cover consisting of clopen sets has a finite subcover) and CLP-compact topological groups. In particular, we extend a theorem on CLP-compactness of products from [J. Steprāns, A. Šostak, Restricted compactness properties and their preservation under products, Topology Appl. 101 (3) (2000) 213-229] and we offer various criteria for CLP-compactness for spaces and topological groups, that work particularly well for precompact groups. This allows us to show that arbitrary products of CLP-compact pseudocompact groups are CLP-compact. For every natural n we construct: (i) a totally disconnected, n-dimensional, pseudocompact CLP-compact group; and (ii) a hereditarily disconnected, n-dimensional, totally minimal, CLP-compact group that can be chosen to be either separable metrizable or pseudocompact (a Hausdorff group G is totally minimal when all continuous surjective homomorphisms G → H , with a Hausdorff group H , are open).
Topological group criterion for in compact-open-like topologies, II
Topology and its Applications, 2009
Topological group Cech-Stone compactification Epimorphism Monomorphism Epi-topology Compact-open topology Compact-zero topology Space with filter Frame Lattice-ordered group Pressing down Aronszajn tree We address questions of when (C(X), +) is a topological group in some topologies which are meets of systems of compact-open topologies from certain dense subsets of X. These topologies have arisen from the theory of epimorphisms in lattice-ordered groups (in this context called "epi-topology"). A basic necessary and sufficient condition is developed, which at least yields enough insight to provide the general answer "sometimes Yes and sometimes No". After some reduction the situation seems to become Set Theory (which view will be reinforced by a sequel to this paper "Topological group criterion for C (X) in compact-open-like topologies, II").
Weakly metrizable pseudocompact groups
Applied General Topology, 2006
We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m > 0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G admits no proper pseudocompact refinement). We introduce also weakly extremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this "symmetric" version of extremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups.
A-paracompactness and Strongly A-screenability in Topological Groups
European Journal of Pure and Applied Mathematics
A space is said to be strongly A-screenable if there exists a σ-discrete refinement for each open cover. In this article, we have investigated some of the features of A-paracompact and strongly A-screenable spaces in topological and semi topological groups. We predominantly show that (i) Topological direct product of (countably) A-paracompact topological group and a compact topological group is (countably) A-paracompact topological group. (ii) All the left and right cosets of a strongly A-screenable subset H of a semi topological group (G, ∗, τ ) are strongly A-creenable.
2006
In Section 1 we prove several classical isomorphism theorems for topological groups. Furthermore, we state sufficient criteria for a topological group to be isomorphic to an inner direct product. In order to do so, we will need an open mapping theorem for topological groups which yields that every surjective morphism between topological groups is open if the groups satisfy certain compactness properties. We proceed in Section 2 by analyzing the structure of certain locally compact groups based on their subgroups. Weil's Lemma consists of two structure results for locally compact Hausdorff groups G. In particular, for each g ∈ G the cyclic group 〈g〉 is either discrete and infinite or has compact closure in G. We continue by classifying certain Abelian topological groups as direct products of a free Abelian group with an open subgroup. Additionally, we state an existence criterion for discrete subgroups of locally compact Abelian Hausdorff groups. Finally, we give some results of purely algebraic nature. This treatise was prepared for the seminar "Locally Compact Groups" held by PD. Dr. Ralf Gramlich in August 2010 at TU Darmstadt. The seminar was structured according to Markus Stroppel's book [3]. Further resources are provided under http://www3.mathematik.tu-darmstadt.de/index.php?id=84&evsid=23&&evsver=880\. Notation The mappings will be denoted as actions from the right. The image of a point x under a map f will be written x f. Composition of mappings transforms likewisely, i.e. x (f •g) := x f g. 1 Topological Aspects of the Isomorphism Theorems Definition 1.1 (Quotient Map). Let f : X → Y be a surjective map between topological spaces. We call f a quotient map, if it induces the quotient topology on Y. This means every subset U ⊂ Y is open if and only U f −1 ⊂ X is open. Lemma 1.2 (Universal Mapping Property of Quotient Maps). We consider maps between topological spaces h: X → Y and g : Y → Z with f := h• g. If h is a quotient map and f is continuous, then g is also continuous. Hence, we have the following situation: X h , 2 f 8 8 Y g / / Z Proof. Let O ⊂ Z be open. As f is continuous, O f −1 is open. By definition of the mappings O g −1 h −1 = O f −1. As h induces the quotient topology on Y and O g −1 h −1 is open, also O g −1 is open.
A Note on Locally Compact Subsemigroups of Compact Groups
arXiv (Cornell University), 2020
An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group.
Zero-Dimensionality of Some Pseudocompact Groups
Proceedings of the American Mathematical Society, 1994
We prove that hereditarily disconnected countably compact groups are zero-dimensional. This gives a strongly positive answer to a question of Shakhmatov. We show that hereditary or total disconnectedness yields zerodimensionality in various classes of pseudocompact groups.