Hiroki Kodama had a seminar talk in the iTHEMS math seminar on 20 June. The topic was the foundation of metric space and its generalizations. Metric space is one of the most primitive setting of geometry. This notion is defined in an abstract way by metric function satisfying 3 axioms. Not only the Euclidean metric, which is familiar for all of us, many ideas arising all over the science (e.g. the distance of two DNAs) are formulated as a kind of metric. A main interest of this seminar was a generalization of metric space, which is less popular even for mathematicians. A function satisfying the axioms of metric space except for the symmetry axiom d(x,y)=d(y,x) is called an asymmetric metric (a typical example is the cost for a ship on a river to move from one point to another). In the first half of the talk, Kodama introduced the basic notion of (asymmetric) metric spaces with various examples. In the second half, he introduced an interest of asymmetric metric space by showing rich information it includes.