Paratopological and semitopological groups versus topological groups (original) (raw)

Subgroups of paratopological groups and feebly compact groups

Applied General Topology, 2014

It is shown that if all countable subgroups of a semitopological group G are precompact, then G is also precompact and that the closure of an arbitrary subgroup of G is again a subgroup. We present a general method of refining the topology of a given commutative paratopological group G such that the group G with the finer topology, say, σ is again a paratopological group containing a subgroup whose closure in (G, σ) is not a subgroup. It is also proved that a feebly compact paratopological group H is perfectly κ-normal and that every G δ-dense subspace of H is feebly compact.

A characterization of completely regular spaces with applications to paratopological groups

2014

We prove that a semiregular topological space XXX is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.

On reflections and three space properties of semi(para)topological groups

Topology and its Applications, 2017

We show that many topological properties are invariant and/or inverse invariant under taking T 2-reflections in semitopological groups. We also extend some three space properties in topological groups (paratopological groups) to semitopological groups. The following result is established: Let G be a regular semitopological group and let H be a closed subgroup of G such that all compact (resp., countably compact, sequentially compact) subsets of the semitopological group H are first-countable. If the quotient space G/H has the following property, then so does the semitopological group G. (*) all compact (resp., countably compact, sequentially compact) subsets are Hausdorff and strongly Fréchet (strictly Fréchet).

Feebly compact paratopological groups and real-valued functions

Monatshefte für Mathematik, 2012

We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with continuous multiplication) which provide answers to a number of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group G can fail to be a topological group. Our group G has the Baire property, is Fréchet-Urysohn, but it is not precompact. It is well known that every infinite pseudocompact topological group contains a countable non-closed subset. We construct an infinite feebly compact Hausdorff paratopological group G all countable subsets of which are closed. Another peculiarity of the group G is that it contains a nonempty open subsemigroup C such that C −1 is closed and discrete, i.e., the inversion in G is extremely discontinuous. We also prove that for every continuous real-valued function g on a feebly compact paratopological group G, one can find a continuous homomorphism ϕ of G onto a second countable Hausdorff topological group H and a continuous real-valued function h on H such that g = h • ϕ. In particular, every feebly compact paratopological group is R 3-factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups.

Embedding paratopological groups into topological products

Topology and its Applications, 2009

We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that V ∈γ V V −1 ⊆ U. Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the index of regularity of G is countable. As a by-product, we show that a regular totally ω-narrow paratopological group with countable index of regularity is Tychonoff.

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.

On subgroups of saturated or totally bounded paratopological groups

2010

A paratopological group G is saturated if the inverse U −1 of each non-empty set U ⊂ G has non-empty interior. It is shown that a [first-countable] paratopological group H is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if H admits a continuous bijective homomorphism onto a (totally bounded) [abelian] topological group G [such that for each neighborhood U ⊂ H of the unit e there is a closed subset F ⊂ G with e ∈ h −1 (F) ⊂ U ]. As an application we construct a paratopological group whose character exceeds its π-weight as well as the character of its group reflexion. Also we present several examples of (para)topological groups which are subgroups of totally bounded paratopological groups but fail to be subgroups of regular totally bounded paratopological groups.

Each regular paratopological group is completely regular

Proceedings of the American Mathematical Society, 2016

We prove that a semiregular topological space X X is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.

Subgroups of products of certain paratopological (semitopological) groups

Topology and its Applications, 2018

In the first part of this note, we give some sufficient conditions under which a paratopological group is topologically isomorphic to a subgroup of a product of strongly metrizable paratopological groups. In the second part of this note, we show that a regular (Hausdorff, T 1) semitopological group G admits a homeomorphic embedding as a subgroup into a product of regular (Hausdorff, T 1) first-countable semitopological groups which are σ-spaces if and only if G is locally ω-good, ω-balanced, Ir(G) ≤ ω (Hs(G) ≤ ω, Sm(G) ≤ ω) and with the property that for every open neighborhood U of the identity e of G the cover {xU : x ∈ G} has a basic refinement F which is σ-discrete with respect to a countable family V of open neighborhoods of e. In the last part of this note, we give an internal characterization of projectively T i secondcountable semitopological groups, for i = 0, 1, 2.