Symmetrically and Antisymmetrically-dense subspaces of $T_0$-quasi-metric spaces
Keywords:
symmetrization metric, antisymmetrically connected, complementary graph, symmetry component, $T_0$-quasi-metric, symmetrically-dense, antisymmetric pair, symmetric pathAbstract
The aim of this study is to determine some new types of density specific to $T_0$-quasi-metric subspaces in the asymmetric environment due to the apparent inadequacy of topological density in the transfer of properties symmetric and antisymmetric connectedness to the subspaces or superspaces. By taking into consideration the deficiencies of topological density in the relevant theories, the authors introduced and observed the two special kinds of density in the context of non-metric $T_0$-quasi-metrics under the names symmetric density and antisymmetric density.
In this context, some asymmetric properties of these types of density, as well as their various characterizations and relations with the topological density are investigated by presenting many crucial results, (counter)examples and metric approaches. More specifically, we address the following questions: under which conditions does a $T_0$-quasi-metric space become symmetrically or antisymmetrically connected whenever it has a symmetrically or antisymmetrically connected subspace, respectively? Hence, our primary interest is to determine the corresponding responses to these questions.
References
J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, NorthHolland, New York, Fifth Printing. 1982.
N. Demetriou and H.-P.A. Künzi, A study of quasi-pseudometrics, Hacet. J. Math. Stat. 46 (1) (2017), 33-52.
A. Hellwig and L. Volkmann, The connectivity of a graph and its complement, Appl. Math., 156 (2008), 3325-3328.
N. Javanshir and F. Yıldız, Symmetrically connected and antisymmetrically connected $T_0$-quasi-metric extensions, Topol. Appl, 276 (2020), 107179.
H.-P.A. Künzi, An introduction to quasi-uniform spaces, in: Beyond Topology, eds. F. Mynard and E. Pearl (Eds.), Beyond Topology, in: Contemp. Math., 486 (2009), AMS, 239-304.
H.-P. A. Künzi and F. Yıldız, Convexity structures in $T_0$-quasi-metric spaces, Topol. Appl., 200 (2016), 2-18.
H.-P. A. Künzi and F. Yıldız, Extensions of $T_0$-quasi-metrics, Acta Math. Hungar., 153 (1) (2017), 196-215.
F.M. Solofomananirina Tiantsoa and H.-P.A. Künzi, Splitting ultra-metrics by $T_0$-ultra-quasi-metrics, Topol. Appl, 240 (2018), 21-34.
R. J. Wilson, Introduction To Graph Theory, Oliver and Boyd, Edinburgh, 1972.
F. Yıldız and H.-P. A. Künzi, Symmetric Connectedness in $T_0$-quasi-metric spaces, Bull. Belg. Math. Soc. Simon Stevin, 26 (5) (2019), 659-679.
