About one quasi-metric space

The ${M}$-space $(X, \rho)$ is defined as a non-empty set $X$ with distance $\rho: X^2 \to \mathbb {R}_+$ satisfying the axiom of identity and the weakened triangle inequality. The ${M}$-space $(X, \rho)$ belongs to the class of $f$-quasi-metric spaces, and the map $\rho$ may not be $(c_1, c_2)$-quasi-metric for any values of $c_1, \, c_2;$ and $(c_1, c_2)$-quasi-metric space may not be an ${M}$-space. The properties of the ${M}$-space are investigated. An extension of the Krasnosel'skii theorem about a fixed point of a generally contracting map to the ${M}$-space is obtained.