tags:definition |topology

Metric

Context

Statements

Definition. (Citation Needed) Let $X$ be a set, a function $d:X\times X\to \mathbb{R}$ is said to be a metric if

• It is symmetric: for every $x_1,x_2\in X$, the equality $d(x_1,x_2)=d(x_2,x_1)$ holds.
• Indicernible: for every $x_1,x_2\in X$ such that $x_1=x_2$, the equality $d(x_1,x_2)=0$ holds.
• Positive: for every $x_1,x_2\in X$ such that $x_1\ne x_2$, the inequality $d(x_1,x_2)>0$ holds.
• Subadditive: for every $x_1,x_2,x_3\in X$, the inequality $d(x_1,x_3)\le d(x_1,x_2)+d(x_2,x_3)$ holds.