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We're asked to show that the following mappings [itex]d_i:M_i\times{M_i}\rightarrow{\mathbb{R}}[/itex] are metrics on the sets [itex]M_i[/itex]. Moreover, we should think about what the corresponding open neighbourhood [itex]B_{\epsilon}(x)=\{y\in{M_i}|d_i(x,y)<\epsilon\}[/itex] for a point [itex]x\in{M_i}[/itex] and [itex]\epsilon>0[/itex] looks like.

1) Discrete metric: for a set [itex]M_1[/itex]

[itex]d_1:M_1\times{M_1}\rightarrow{\mathbb{R}},\displaystyle d_1(x,y)=\begin{cases}0&\text{if }x=y\\1&\text{otherwise} \end{cases}.[/itex]

2) The French Railway Metric: For [itex]M_2={\mathbb{R}}^2[/itex]

[itex]d_2:M_2\times{M_2}\rightarrow{\mathbb{R}},\displaystyle d_2(x,y)=\begin{cases}||x-y||&\text{if }y=tx(t\in{\mathbb{R}})\\||x||+||y||&\text{otherwise} \end{cases}.[/itex]

3) p-adic Metric on [itex]\mathbb{Z}[/itex]

Let [itex]M_3=\mathbb{Z}[/itex] and p is a prime number:

[itex]d_3:M_3\times{M_3}\rightarrow{\mathbb{R}},d_3(a,b)=inf\{2^{-n}|n\in{\mathbb{N}};p^n\\divides\\|a-b|\}[/itex]

Please, give me at least a hint about what the structure of the proof might be...how to start and how to end.

...and tell me is it normal if you do such things in the first three weeks of your undergraduate maths study?...I mean are those concepts something that you should "mellow" to be able to accumulate and understand clearly?

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# Homework Help: Help with metrics

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