On fixed points of contraction maps acting in $(q_1, q_2)$-quasimetric spaces and geometric properties of these spaces

We study geometric properties of $(q_1, q_2)$-quasimetric spaces and fixed point theorems in these spaces. In paper [1], a fixed point theorem was obtained for a contraction map acting in a complete $(q_1, q_2)$-quasimetric space. The graph of the map was assumed to be closed. In this paper, we show that this assumption is essential, i.e. we provide an example of a complete quasimetric space and a contraction map acting in it whose graph is not closed and which is fixed-point-free. We also describe some geometric properties of such spaces.