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?