Show the "clock"and Euclidean metrics generate different topologies I'm trying to teach my self topology. I wanted to find an example of a metric generating different topology. I came up with what a call "clock" metric, inspired by the modulo operation.  Can anyone please look this over and verify that my reasoning is correct. Also, does this type of metric have a standard name, and/or generalization to higher dimensions?
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Clock metric: 
$$d(x,y)=6-||x-y|-6|$$
The distance between two numbers on a clock. Ex $d(2,11)=3$ (see diagram)



Show the clock metric(above) and the Euclidean metric generate different topologies on the set [1,13).

Consider the open ball $B_{2}^{clock}(1)=(11,13) \cup [1,3)$. There does not exist $0<r<2$, such that $\{12\}\in B_{r}^{Eucl}(1)$.Since $r$ does not exist, the topologies generated must be different. 
 A: The space $\langle[0,13),d\rangle$ is homeomorphic to the space $S^1$, the unit circle in the plane, with its Euclidean metric. Indeed, your metric is equivalent to that metric. This means that your space is compact; since $[0,13)$ with the Euclidean metric is not compact, it is quite true that your metric gives $[0,13)$ a different topology.
A: (one year later) I think I have a valid proof now:

(1). The def'n of the topology $T_d$ generated by a metric $d$ on a set $S$ is the topology whose base is the set of open $d$-balls. 
  (2). Two  metrics $d,d'$ on $S$ are inequivalent (meaning that $T_d\ne T_{d'}$) iff there exists $U\subset S$ which belongs to one of $T_d,T_{d'}$ but not the other.

Without loss of generality , suppose $U\in T_d$  \  $T_{d'}. $ Then, by (1), every $p\in U$ belongs to an open $d$-ball that is a subset of $U, $ but there must exist some $p\in U$ such that  no open $d'$-ball containing $p$ is a subset of $U.$ In particular $B_{d'}(p,1/n) \not \subset U$ for every $n\in \mathbb Z^+.$
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note: I'm using the open ball notation: $B_{radius}^{metric}(point)$
 ex) $B_{1}^{Eucl}(2)$, is a Euclidean ball of radius 1, centered at 2.
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Let $S = [1,13)$; d = Euclidean metric; d' = "Clock" metric. Some examples of $U$'s:
$U_1 =B_{1}^{clock}(2)=B_{1}^{Eucl}(2)= (1,3)$ ; $U_1 \in T_d \cap T_{d'}$
$U_2 =B_{2}^{clock}(1) = B_{1}^{Eucl}(12) \cup B_{2}^{Eucl}(1) = (11,13) \cup [1,3)$ ; $U_2 \in T_d \cap T_{d'}$
$U_3 = B_{2}^{Eucl}(1) = [1,3)$ ; $U_3 \in T_d ; U_3 \notin  T_{d'} $ 
 Thus,  $U_3\in T_d$  \  $T_{d'}. $ 
 $U_1$ shows that some balls are the same in either metric. $U_2$ shows a ball in the clock metric, that is actually two balls in the Euclidean metric. $U_3$ show a subset that is open in Euclidean, but closed in Clock. 
In particular $B_{1/n}^{clock}(1) \not \subset U$ for every $n\in \mathbb Z^+.$
