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Following the theory of symmetrically connected  T0 -quasi-metric spaces constructed lately, in the present investigation the authors introduce and study the localized version of symmetric connectedness under the name local symmetric connectedness, as a new approach to determining the degree of the asymmetry and symmetry of  T0 -quasi-metric spaces. In this context, some properties and topological aspects of locally symmetrically connected  T0 -quasi-metric spaces are discussed by presenting various observations and many concrete (counter)examples. Specifically, it is shown that symmetric connectedness coincides with the local symmetric connectedness if we deal with the  T0 -quasi-metric induced by an asymmetric norm which associates the theory of quasi-metrics in asymmetric topology with functional analysis.

Locally symmetrically connected T 0 -quasi-metric spaces