Within the framework of asymmetry of the $T_0$-quasi-metric spaces [1], antisymmetric functions are appeared [3] as in some sense opposite to metric functions and studied [4] in detail. Following that in a previous study [2], the locality status of the $T_0$-quasi-metric spaces constructed with antisymmetric functions is described under the name local antisymmetricness.

Hence, we are now in a position to ask that how local antisymmetric spaces behaves for subspaces, finite products and intersections-unions. Accordingly, some theorems and counterexamples will be presented about these observations in the context of $T_0$-quasi-metric spaces. Specifically, the question whether the images of locally antisymmetric spaces under an isometry have the same property or not, will be discussed as another problem worth examining.

References

[1] H.-P.A. Künzi, {\it An introduction to quasi-uniform spaces}, in: Beyond Topology, eds. F. Mynard and E. Pearl, Contemporary Mathematics, American Mathematical Society, 486, 239–304, 2009.

[2] F. Yıldız, Localization of Antisymmetric Spaces in the Framework of Quasi-Metrics, {\em Topology Proceedings}, Vol. 66, 185–200, 2025.

[3] F. Yıldız and H.-P. A. Künzi, Symmetric Connectedness in $T_0$-quasi-metric spaces, {\em Bull. Belg. Math. Soc. Simon Stevin}, Vol. 26 \ (5) 659–679, 2019.

[4] F. Yıldız and N. Javanshir , Certain observations on antisymmetric $T_0$-quasi-metric spaces, {\em Topology and Its Applications}, 309, 107915, 2022.