Some topological properties of $f$-quasimetric~spaces

The properties of the $f$–quasimetric space $(X,\rho)$ are studied. In a space as such, the distance $\rho$  satisfies the identity axiom and the generalized triangle inequality: $\rho(x,z) \leq f(\rho(x,y),\rho(y,z))$ for any $x,y,z\in X.$ Here the function $f$ is positive for positive arguments, continuous at the point $(0,0)$, and $f(0,0)=0.$ The symmetry of the distance is not assumed. The topology on $X$ generated by the distance $\rho$ is defined in the standard way. The properties of convergent sequences and sequentially compact sets are studied. Conditions are obtained under which the convergence in itself (mutual convergence) is necessary for the convergence of a sequence. The relationship between the rates of convergence of a fundamental sequence and its convergence in itself is considered. The concept of a sequentially precompact set is introduced. Conditions are obtained under which the closure of a sequentially precompact set is sequentially compact.