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.