Abstract
This article is devoted to studying the structure of topological left (or right) quasigroups, which play a great role in noncommutative geometry. Quotient and transversal mappings are important in the theory of differentiable manifolds and topological manifolds. Their transversal and quotient mappings are investigated. Necessary and sufficient conditions for their continuity are scrutinized. Examples are given. Homogeneous spaces are investigated related to topological quasigroups and their subquasigroups. For this purpose, the products of special types of topological left (or right) quasigroups, which are called smashed, are investigated. They are used to describe an extensive family of topological nondiscrete left (or right) quasigroups for which transversal mappings are continuous.
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