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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?

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    $\begingroup$ Drop it and see if you can get a topological space. You will get a basis, but then you will not be able to show that open balls are open sets. $\endgroup$ – Moishe Kohan Feb 12 '14 at 9:04
  • $\begingroup$ Perhaps one application can help: how would you show that a plane without some line is an open set? $\endgroup$ – Jonathan Y. Feb 12 '14 at 9:05
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By transitivity of closeness, your book probably means something like this: if $d(x, y) < \epsilon$ and $d(y, z) < \epsilon$, then $d(x, z) \leq d(x, y) + d(y, z) < \epsilon + \epsilon = 2\epsilon$; that is, if $x$ is "close" to $y$ and $y$ is "close" to $z$, then $x$ is "close" to $z$.

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  • $\begingroup$ @Lansiz: This transitivity is essential. It tells you that for the ball $B_{2ϵ}(x)$ the ball $B_ϵ(x)$ is the neighborhood of $x$ of which $B_{2ϵ}(x)$ is a neighborhood. This property "Every neighborhood of $x$ is also a neighborhood for points sufficiently close to $x$" is a substantial part of the definition of a neighborhood system. $\endgroup$ – Stefan Hamcke Feb 12 '14 at 15:07
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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.

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