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X = ( $\omega$, $d_F$), where $d_f(x, y) = \sum_{n=1}^{\infty} 2^{-k}\frac{\lvert x_k-y_k\rvert}{1+\lvert x_k-y_k\rvert}$, and $\omega$ is the set all real sequences. Let $A = \{(x_n): \lvert x_n\rvert \leq 1 , \mbox{for all n} \}$. Find IntA, and $\bar{A}$.

I am trying to understand the concept of this metric. This metric is bounded by 1 obviously. When I compare two sequences it seems to me that only the part $\frac{\lvert x_k-y_k\rvert}{1+\lvert x_k-y_k\rvert}$ is important. But I could not find an intuition for this part. Firstly, can you give me some intuition about this.

Moreover, $f(x)=\frac{x}{1+x}$ is increasing. Thus, if $S = \sup\{ \lvert x_k-y_k\rvert $}, then $\frac{S}{1+S} \geq \frac{\lvert x_k-y_k\rvert}{1+\lvert x_k-y_k\rvert}$ for all k. Therefore, it seems to me that this metric is related to the supremum metric of course provided that the difference is convergent (or whatever we need, I am not quite sure). Basically, I could not find the interior points of this set, since I do not understand the metric.

Can you please help me with this question? Thanks in advance for every comment.

Hanul Jeon
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Mahmut Esat Akın
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  • If you are acquainted with a bit of topology, then you may ask 'what is a more natural description of the topology given by this funky metric $d_F$?'. The answer is that it is exactly the product topology on $\mathbb{R}^{\mathbb{N}}$. Alternatively, one may describe it ans the topology given by the family of pseudometrics $d_k(x, y) = |x_k-y_k|$ for $k = 1, 2, \cdots$. (Here, a pseudometric is a function satisfies all the axioms of metric except that it may partially lack the ability of resolving two different points.) – Sangchul Lee Oct 24 '19 at 18:32
  • Yet another description is that, a sequence $(x^{(n)}){n\geq1}$ in $\mathbb{R}^{\mathbb{N}}$ under $d_F$ converges to $x$ if and only if each component $(x_k^{(n)}){n\geq1}$ converges to $x_k$ for $k = 1, 2, \cdots$. – Sangchul Lee Oct 24 '19 at 18:38

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