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. 

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.
