On Neighborhood Groups
Abstract
In this study, we introduce the notion of a neighborhood group as a generalization of a topological group by formulating the compatibility of the group operations with the neighborhood structure in terms of a closure operator. We prove that every topological group is a neighborhood group and provide examples showing that the converse does not hold. We characterize neighborhood groups through the compatibility of the difference map with the closure operator. We also prove that every subgroup, equipped with the induced closure operator, is a neighborhood group. Furthermore, we prove that left and right translations are homeomorphisms, and inner automorphisms preserve the closure structure. Finally, we investigate separation axioms and show that, unlike in topological groups, a $T_1$ neighborhood group need not be $T_2$. These results demonstrate that several fundamental properties of topological groups extend to algebraic structures equipped with a closure operator, even when the underlying structure is not topological.
Keywords
Topological groups, Neighborhood spaces, Closure operators, Continuity, Separation axioms
References
- F. Klein, S. Lie, Ueber diejenigen ebenen Curven, welche durch ein geschlossenes System von einfach unendlich vielen vertauschbaren linearen Transformationen in sichübergehen, Math. Ann., 4 (1871), 50–84. https://doi.org/10.1007/BF01443297
- F. Klein, Vergleichende Betrachtungen Uber Neuere Geometrische Forschungen, Andreas Deichert, Erlangen, 1872.
- S. Lie, Theorie der Transformationsgruppen I, Math. Ann., 16 (1880), 441–528. https://doi.org/10.1007/BF01446218
- D. Montgomery, What is a topological group?, Amer. Math. Monthly, 52(6) (1945), 302–307.
- P. J. Higgins, An Introduction to Topological Groups, London Math. Soc. Lecture Note Ser., Vol. 15, Cambridge University Press, Cambridge, 1974.
- O. Schreier, Abstrakte kontinuierliche Gruppen, Abh. Math. Semin. Univ. Hambg., 4 (1926), 15–32. https://doi.org/10.1007/BF02950716
- L. S. Pontryagin, Topological Groups, Princeton Mathematical Series, Vol. 2, Princeton University Press, Princeton, 1946.
- L. S. Pontryagin, Selected Works, Vol. 2: Topological Groups, 3rd ed., R. V. Gamkrelidze (ed.), Gordon and Breach Science Publishers, 1986.
- T. Husain, Introduction to Topological Groups, W. B. Saunders Company, Philadelphia, 1966.
- F. Hausdorff, Grundz¨uge der Mengenlehre, Verlag von Veit & Comp., Leipzig, 1914.