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1
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Semi-quotient mappings and spaces

100%
Khan M. u. D., Noreen R., Siddique Bosan M.
Open Mathematics
|
2016
|
tom 14
|
nr 1
1014-1022
EN
In this paper, we continue the study of s-topological and irresolute-topological groups. We define semi-quotient mappings which are stronger than semi-continuous mappings, and then consider semi-quotient spaces and groups. It is proved that for some classes of irresolute-topological groups (G, *, τ) the semi-quotient space G/H is regular. Semi-isomorphisms of s-topological groups are also discussed.
2
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Perfectly supportable semigroups are σ-discrete in each Hausdorff shift-invariant topology

100%
Banakh T., Guran I.
Topological Algebra and its Applications
|
2013
|
tom 1
1-8
EN
In this paper we introduce perfectly supportable semigroups and prove that they are σ-discrete in each Hausdorff shiftinvariant topology. The class of perfectly supportable semigroups includes each semigroup S such that FSym(X) ⊂ S ⊂ FRel(X) where FRel(X) is the semigroup of finitely supported relations on an infinite set X and FSym(X) is the group of finitely supported permutations of X.
3
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Group reflection and precompact paratopological groups

100%
Tkachenko M.
Topological Algebra and its Applications
|
2013
|
tom 1
22-30
EN
We construct a precompact completely regular paratopological Abelian group G of size (2ω)+ such that all subsets of G of cardinality ≤ 2ω are closed. This shows that Protasov’s theorem on non-closed discrete subsets of precompact topological groups cannot be extended to paratopological groups. We also prove that the group reflection of the product of an arbitrary family of paratopological (even semitopological) groups is topologically isomorphic to the product of the group reflections of the factors, and that the group reflection, H*, of a dense subgroup G of a paratopological group G is topologically isomorphic to a subgroup of G*.
4
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Summable Family in a Commutative Group

100%
Coghetto R.
Formalized Mathematics
|
2015
|
tom 23
|
nr 4
279-288
EN
Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.
5
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On the extent of star countable spaces

81%
Alas O., Junqueira L., Mill J., Tkachuk V., Wilson R.
Open Mathematics
|
2011
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tom 9
|
nr 3
603-615
EN
For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω 1-monolithic compact space X, if C p(X)is star countable then it is Lindelöf.
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