TITLE: Why is the triangle inequality property important for metric spaces?
QUESTION [6 upvotes]: There is a property in the axiomatization of metric space: $d(x,z)\le d(x,y)+d(y,z)$. I understand that property applies well when it come to the measure of distances among points. However, that property seems to be irrelevant in the definitions of continuity, open set and some other key concepts.
My topology book says that it is asserting "the transitivity of closeness", a vague term that loses me. So why is that property important? Why did the founders (if any) of topology consider it at the first place? And what if that property is missed?

REPLY [4 votes]: Metric is actually a way of measuring distance between two points in space. The triangular inequality is intuitively a generalization (also a weaker condition) of the fact that the shortest path from one point to the other is the line segment connecting two points. Also, if triangular inequality doesn't hold, one will see something really absurd. A ridiculous example is
$$0=d(x,x)\geq d(x,y)+d(y,x)\geq 0$$
Hence $d(x,y)=0$ for every point $y$, which makes all points coincide from the metric view. In short, triangular inequality promise the definition of metric is well-defined and meaningful.