Give a concrete description of all open balls in $\mathbb{Z}$.

Question

Let $\mathbb{Z}$ be the endowed with the induced metric from $\mathbb{R}$. Give a concrete description of all open balls in $\mathbb{Z}$.

My attempt: Let $x$ be an integer. Then $B(x, r)= \{x\}$ for $r\in (0,1]$, $B(x, r)= \{x-1,x,x+1\}$ for $r\in (0,2]$, $B(x, r)= \{x-2,x-1,x,x+1,x+2\}$ for $r\in (0,3]$, etc.

So, my answer is $B(x, r)= \{x-(k-1), x-(k-2), \cdots, x-1,x,x+1,\cdots, x-(k+1)\}$ for $r\in (0,k]$.

Is it correct?

$r$ may be in $(0,1]$ as well as $(0,2]$! – geetha290krm Jun 29 '22 at 11:56 — geetha290krm, Jun 29 '22 at 11:56

score 1 · Accepted Answer · answered Jun 29 '22 at 11:55

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No, your answer has a contradiction in iteself. You should say $B(x, r)= \{x-(k-1), x-(k-2), \cdots, x-1,x,x+1,\cdots, x+k-1\}$ for $r\in (k-1,k]$.

answered Jun 29 '22 at 11:55

geetha290krm

36,632

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Thanks for answer. I have understood my mistake. – user1234 Jun 29 '22 at 12:02

Give a concrete description of all open balls in $\mathbb{Z}$.

1 Answers1