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Let $\left\{1,2,3\right\}$ be equipped with the discrete topology and $X=\left\{1,2,3\right\}^{\mathbb{Z}}$ with the product-topology. Then one possible metric on $X$ is $$ d(x,y)=\begin{cases}2^{-k} \text{ with k maximal such that }x_{[-k,k]}=y_{[-k,k]}, & x\neq y\\0, & x=y \end{cases}. $$

Now let $x\neq y$. What is then $d(x,y)$ if there is no such $k$?

Rhjg
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  • in this case $x=y$ – Euler88 ... Aug 02 '15 at 09:53
  • Why that? I do not see that. – Rhjg Aug 02 '15 at 09:57
  • Look at $x=...231231~ 2~ 31231...$ and $y=...123123~1~23123...$, where the spaces mark the zero position. They are not equal but there is no such k. – Rhjg Aug 02 '15 at 10:00
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    If $x_0=y_0$, but $x_{[-1,1]}\ne y_{[-1,1]}$ then $d(x,y)=2^{-0}=1$, according to your formula. If even $x_0\ne y_0$ I'd assume that $d(x,y)=2$ is meant. – Christian Blatter Aug 02 '15 at 19:12
  • You mean: $d(x,y)=\begin{cases}2, & x\neq y, x_0\neq y_0\2^{-k}\text{ with k maximal such that }x_{[-k,k]}=y_{[-k,k]}, & x\neq y, x_0=y_0\0, & x=y\end{cases}$? Why 2 in the first case? Can't I choose any value >1? – Rhjg Aug 04 '15 at 19:22
  • You know, I'd rather agree with @ChristianBlatter now. When you put $k$ equal to -1, you encode an empty interval $[1, -1]$ which nicely describes the case $x \neq y$, $x_0 \neq y_0$. – Evgeny Aug 06 '15 at 17:55

1 Answers1

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In symbolic dynamics people use similar metrics for spaces of sequences and the motivation for the formulas is always the following: "the longer central block on which two sequences coincide, the closer they are to each other". I think in your undefined case sequences $x$ and $y$ "coincide on block of zero length", so it's tempting to say that you assign distance $d(x, y) = 2^{-0} = 1$ to them (while it's tempting, it's not so true to do it this way, see an addition).

ADDED LATER:
There's a presentation by Schlomo Sternberg, take a look at slide 7. He defines the distance between sequences exactly the same way as in your question. He also notes that convention $\lbrack x_k , x_l \rbrack$ denotes an empty block if $k > l$. Of course, any $\lbrack x_{-i}, x_{+i} \rbrack$ with $i > 0$ denotes an empty block, the "minimal" empty block (in terms of maximal $i$) is $\lbrack x_{1}, x_{-1} \rbrack$. So, the Cristian Blatter's suggestion for $d(x, y) =2 $ when $x \neq y$ seems to be most reasonable.

Evgeny
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  • Up to my definition, the value 1 is reserved for the case that $x\neq y, x_0=y_0, x_{[-1,1]}\neq y_{[-1,1]}$. So is it really good to put the value 1, too, if $x\neq y, x_0\neq y_0$? – Rhjg Aug 04 '15 at 19:07