Various results on some asymmetric types of density

Year 2023, Volume: 72 Issue: 4, 1173 - 1186, 29.12.2023

Nezakat Javanshır , Filiz Yıldız

https://doi.org/10.31801/cfsuasmas.1285637

Abstract

The structures of symmetric connectedness and dually, antisymmetric connectedness were described and studied before, especially in terms of graph theory as the corresponding counterparts of the connectedness of a graph and the connectedness of its complementary graph. By taking into consideration the deficiencies of topological density in the context of symmetric and antisymmetric connectedness, two special kinds of density in the theory of non-metric $T_0$-quasi-metrics were introduced in the previous studies under the names symmetric density and antisymmetric density. In this paper, some crucial and useful properties of these two types of density are investigated with the help of the major results and (counter)examples peculiar to the asymmetric environment. Besides these, many further observations about the structures of symmetric and antisymmetric-density are dealt with, especially in the sense of their combinations such as products and unions through various theorems in the context of $T_0$-quasi-metrics. Also, we examine the question of under what kind of quasi-metric mapping these structures will be preserved.

Keywords

T_0-quasi-metric, connected graph, symmetrization topology, antisymmetric point, Star space, symmetrically-dense, antisymmetrically connected, complementary graph, isometric isomorphism, symmetric path

References

• Bondy, J.A., Murty, U.S.R., Graph Theory with Applications, North-Holland, New York, Fifth Printing. 1982.
• Cobzaş, Ş., Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Springer, Basel, 2013.
• Demetriou, N., Künzi, H.-P.A., A study of quasi-pseudometrics, Hacet. J. Math. Stat., 46 (1) (2017), 33-52. https//doi.org/10.15672/HJMS.2016.396.
• Hellwig, A., Volkmann, L., The connectivity of a graph and its complement, Appl. Math., 156 (2008), 3325-3328. https//doi.org/10.1016/j.dam.2008.05.012.
• Javanshir, N., Yıldız, F., Symmetrically connected and antisymmetrically connected $T_0$-quasi-metric extensions, Top. and Its Appl., 276 (2020), 107179. https://doi.org/10.1016/j.topol.2020.107179.
• Javanshir, N., Yıldız, F., Locally symmetrically connected $T_0$-quasi-metric spaces, Quaest. Math., 45 (3) (2022), 369-384. https://doi.org/10.2989/16073606.2021.1882602.
• Künzi, H.-P.A., An introduction to quasi-uniform spaces, in: Beyond Topology, eds. F. Mynard and E. Pearl (Eds.), Beyond Topology, in: Contemp. Math., 486 (2009), 239-304.
• Künzi, H.-P. A., Yıldız, F., Extensions of T0-quasi-metrics, Acta Math. Hungar., 153 (1) (2017), 196-215. https://doi.org/10.1007/s10474-017-0753-z.
• Künzi, H.-P.A., Yıldız, F., Javanshir, N., Symmetrically and antisymmetrically-dense subspaces of $T_0$-quasi-metric spaces, Top. Proc., 61 (2023), 215-231.
• Wilson, R. J., Introduction To Graph Theory, Oliver and Boyd, Edinburgh, 1972.
• Yıldız, F., Künzi, H.-P.A., Symmetric connectedness in $T_0$-quasi-metric spaces, Bull. Belg. Math. Soc. Simon Stevin, 26 (5) (2019), 659-679. https://doi.org/10.36045/bbms/1579402816.
• Yıldız, F., Javanshir, N., On the topological locality of antisymmetric connectedness, Filomat, 37 (12) (2023), 3879-3886. https://doi.org/10.2298/FIL2312883Y.